  
  [1X50 [33X[0;0YGroup Libraries[133X[101X
  
  [33X[0;0YWhen you start [5XGAP[105X, it already knows several groups. Currently [5XGAP[105X initially
  knows the following groups:[133X
  
  [30X    [33X[0;6Ysome   basic  groups,  such  as  cyclic  groups  or  symmetric  groups
        (see [14X50.1[114X),[133X
  
  [30X    [33X[0;6YClassical matrix groups (see [14X50.2[114X),[133X
  
  [30X    [33X[0;6Ythe  transitive  permutation  groups of degree at most 30, provided by
        the [5XTransGrp[105X package (see [14X'transgrp: Transitive Permutation Groups'[114X),[133X
  
  [30X    [33X[0;6Ya  library  of  groups of small order (see [14X'smallgrp: The Small Groups
        Library'[114X),[133X
  
  [30X    [33X[0;6Ythe finite perfect groups of size at most [22X2⋅ 10^6[122X (see [14X50.6[114X).[133X
  
  [30X    [33X[0;6Ythe  primitive  permutation  groups  of degree [22X< 4096[122X, provided by the
        [5XPrimGrp[105X package (see [14X'primgrp: Primitive Permutation Groups'[114X),[133X
  
  [30X    [33X[0;6Ythe  irreducible  solvable subgroups of [22XGL(n,p)[122X for [22Xn>1[122X and [22Xp^n < 256[122X,
        provided  by  the  [5XPrimGrp[105X package (see [14X'primgrp: Irreducible Solvable
        Matrix Groups'[114X),[133X
  
  [30X    [33X[0;6Ythe  irreducible maximal finite integral matrix groups of dimension at
        most 31 (see [14X50.7[114X),[133X
  
  [30X    [33X[0;6Ythe crystallographic groups of dimension at most 4[133X
  
  [33X[0;0YThere  is  usually no relation between the groups in the different libraries
  and a group may occur in different libraries in different incarnations.[133X
  
  [33X[0;0YNote  that  a system administrator may choose to install all, or only a few,
  or  even none of the libraries. So some of the libraries mentioned below may
  not be available on your installation.[133X
  
  [33X[0;0Y[5XGAP[105X  might  use  data libraries that are available to speed up calculations,
  for  example  in  using  a  classification  to determine that groups must be
  isomorphic,  based  on  agreement  of  properties;  or  to determine maximal
  subgroups  or subgroup maximality. This will be indicated by an info message
  of  level  2  in the info class [10XInfoPerformance[110X. If the calculation is to be
  independent  of  such  data library use, for example if it is used to verify
  the data library, functions can be called with the option [10XNoPrecomputedData[110X,
  to  turn  these  features  off.  Doing  so  might cause significantly longer
  calculations, or even failure of certain calculations.[133X
  
  
  [1X50.1 [33X[0;0YBasic Groups[133X[101X
  
  [33X[0;0YThere  are  several  infinite  families  of groups which are parametrized by
  numbers.  [5XGAP[105X  provides  various  functions  to  construct these groups. The
  functions  always  permit  (but  do  not  require)  one to indicate a filter
  (see [14X13.2[114X),  for  example  [2XIsPermGroup[102X  ([14X43.1-1[114X),  [2XIsMatrixGroup[102X ([14X44.1-1[114X) or
  [2XIsPcGroup[102X ([14X46.3-1[114X), in which the group shall be constructed. There always is
  a  default  filter  corresponding  to a [21Xnatural[121X way to describe the group in
  question.  Note  that  not  every  group can be constructed in every filter,
  there  may  be  theoretical  restrictions ([2XIsPcGroup[102X ([14X46.3-1[114X) only works for
  solvable groups) or methods may be available only for a few filters.[133X
  
  [33X[0;0YCertain  filters may admit additional hints. For example, groups constructed
  in  [2XIsMatrixGroup[102X  ([14X44.1-1[114X) may be constructed over a specified field, which
  can  be  given as second argument of the function that constructs the group;
  The default field is [2XRationals[102X ([14X17.1-1[114X).[133X
  
  [1X50.1-1 TrivialGroup[101X
  
  [33X[1;0Y[29X[2XTrivialGroup[102X( [[3Xfilter[103X] ) [32X function[133X
  
  [33X[0;0Yconstructs  a  trivial  group in the category given by the filter [3Xfilter[103X. If
  [3Xfilter[103X  is not given it defaults to [2XIsPcGroup[102X ([14X46.3-1[114X). For more information
  on possible values of [3Xfilt[103X see section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XTrivialGroup();[127X[104X
    [4X[28X<pc group of size 1 with 0 generators>[128X[104X
    [4X[25Xgap>[125X [27XTrivialGroup( IsPermGroup );[127X[104X
    [4X[28XGroup(())[128X[104X
  [4X[32X[104X
  
  [1X50.1-2 CyclicGroup[101X
  
  [33X[1;0Y[29X[2XCyclicGroup[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  the  cyclic  group of size [3Xn[103X in the category given by the filter
  [3Xfilt[103X.  If  [3Xfilt[103X  is  not  given  it defaults to [2XIsPcGroup[102X ([14X46.3-1[114X), unless [3Xn[103X
  equals  [2Xinfinity[102X  ([14X18.2-1[114X),  in which case the default filter is switched to
  [2XIsFpGroup[102X  ([14X47.1-2[114X).  For  more  information  on possible values of [3Xfilt[103X see
  section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XCyclicGroup(12);[127X[104X
    [4X[28X<pc group of size 12 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XCyclicGroup(infinity);[127X[104X
    [4X[28X<free group on the generators [ a ]>[128X[104X
    [4X[25Xgap>[125X [27XCyclicGroup(IsPermGroup,12);[127X[104X
    [4X[28XGroup([ (1,2,3,4,5,6,7,8,9,10,11,12) ])[128X[104X
    [4X[25Xgap>[125X [27Xmatgrp1:= CyclicGroup( IsMatrixGroup, 12 );[127X[104X
    [4X[28X<matrix group of size 12 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XFieldOfMatrixGroup( matgrp1 );[127X[104X
    [4X[28XRationals[128X[104X
    [4X[25Xgap>[125X [27Xmatgrp2:= CyclicGroup( IsMatrixGroup, GF(2), 12 );[127X[104X
    [4X[28X<matrix group of size 12 with 1 generator>[128X[104X
    [4X[25Xgap>[125X [27XFieldOfMatrixGroup( matgrp2 );[127X[104X
    [4X[28XGF(2)[128X[104X
  [4X[32X[104X
  
  [1X50.1-3 AbelianGroup[101X
  
  [33X[1;0Y[29X[2XAbelianGroup[102X( [[3Xfilt[103X, ][3Xints[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  an  abelian group in the category given by the filter [3Xfilt[103X which
  is  of isomorphism type [22XC_{[3Xints[103X[1]} × C_{[3Xints[103X[2]} × ... × C_{[3Xints[103X[n]}[122X, where
  [3Xints[103X  must  be a list of non-negative integers or [2Xinfinity[102X ([14X18.2-1[114X); for the
  latter  value  or  0,  [22XC_{[3Xints[103X[i]}[122X  is  taken  as  an infinite cyclic group,
  otherwise  as  a  cyclic  group  of  order  [3Xints[103X[i]. If [3Xfilt[103X is not given it
  defaults  to  [2XIsPcGroup[102X  ([14X46.3-1[114X),  unless any 0 or [10Xinfinity[110X is contained in
  [3Xints[103X,  in  which  the  default filter is switched to [2XIsFpGroup[102X ([14X47.1-2[114X). The
  generators  of  the  group  returned  are  the elements corresponding to the
  factors  [22XC_{[3Xints[103X[i]}[122X and hence the integers in [3Xints[103X. For more information on
  possible values of [3Xfilt[103X see section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAbelianGroup([1,2,3]);[127X[104X
    [4X[28X<pc group of size 6 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XG:=AbelianGroup([0,3]);[127X[104X
    [4X[28X<fp group of size infinity on the generators [ f1, f2 ]>[128X[104X
    [4X[25Xgap>[125X [27XAbelianInvariants(G);[127X[104X
    [4X[28X[ 0, 3 ][128X[104X
  [4X[32X[104X
  
  [1X50.1-4 ElementaryAbelianGroup[101X
  
  [33X[1;0Y[29X[2XElementaryAbelianGroup[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  the  elementary abelian group of size [3Xn[103X in the category given by
  the filter [3Xfilt[103X. If [3Xfilt[103X is not given it defaults to [2XIsPcGroup[102X ([14X46.3-1[114X). For
  more information on possible values of [3Xfilt[103X see section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XElementaryAbelianGroup(8192);[127X[104X
    [4X[28X<pc group of size 8192 with 13 generators>[128X[104X
  [4X[32X[104X
  
  [1X50.1-5 FreeAbelianGroup[101X
  
  [33X[1;0Y[29X[2XFreeAbelianGroup[102X( [[3Xfilt[103X, ][3Xrank[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  the  free  abelian  group of rank [3Xn[103X in the category given by the
  filter  [3Xfilt[103X.  If  [3Xfilt[103X  is not given it defaults to [2XIsFpGroup[102X ([14X47.1-2[114X). For
  more information on possible values of [3Xfilt[103X see section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XFreeAbelianGroup(4);[127X[104X
    [4X[28X<fp group of size infinity on the generators [ f1, f2, f3, f4 ]>[128X[104X
  [4X[32X[104X
  
  [1X50.1-6 DihedralGroup[101X
  
  [33X[1;0Y[29X[2XDihedralGroup[102X( [[3Xfilt[103X, ][3Xn[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  the dihedral group of size [3Xn[103X in the category given by the filter
  [3Xfilt[103X.  If  [3Xfilt[103X  is  not  given  it defaults to [2XIsPcGroup[102X ([14X46.3-1[114X), unless [3Xn[103X
  equals  [2Xinfinity[102X  ([14X18.2-1[114X),  in which case the default filter is switched to
  [2XIsFpGroup[102X  ([14X47.1-2[114X).  For  more  information  on possible values of [3Xfilt[103X see
  section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDihedralGroup(8);[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XDihedralGroup( IsPermGroup, 8 );[127X[104X
    [4X[28XGroup([ (1,2,3,4), (2,4) ])[128X[104X
    [4X[25Xgap>[125X [27XDihedralGroup(infinity);[127X[104X
    [4X[28X<fp group of size infinity on the generators [ r, s ]>[128X[104X
  [4X[32X[104X
  
  [1X50.1-7 IsDihedralGroup[101X
  
  [33X[1;0Y[29X[2XIsDihedralGroup[102X( [3XG[103X ) [32X property[133X
  [33X[1;0Y[29X[2XDihedralGenerators[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XIsDihedralGroup[102X indicates whether the group [3XG[103X is a dihedral group. If it is,
  methods may set the attribute [2XDihedralGenerators[102X to [[3Xt[103X,[3Xs[103X], where [3Xt[103X and [3Xs[103X are
  two elements such that [3XG[103X = [22X⟨ t, s | t^2 = s^n = 1, s^t = s^-1 ⟩[122X.[133X
  
  [1X50.1-8 DicyclicGroup[101X
  
  [33X[1;0Y[29X[2XDicyclicGroup[102X( [[3Xfilt[103X, [[3Xfield[103X, ]][3Xn[103X ) [32X function[133X
  [33X[1;0Y[29X[2XQuaternionGroup[102X( [[3Xfilt[103X, [[3Xfield[103X, ]][3Xn[103X ) [32X function[133X
  
  [33X[0;0Y[2XDicyclicGroup[102X  constructs the dicyclic group of size [3Xn[103X in the category given
  by  the  filter  [3Xfilt[103X.  Here,  [3Xn[103X  must  be  a  multiple  of  4.  The synonym
  [2XQuaternionGroup[102X  for  [2XDicyclicGroup[102X  is provided for backward compatibility,
  but  will  print a warning if [3Xn[103X is not a power of [22X2[122X. If [3Xfilt[103X is not given it
  defaults  to  [2XIsPcGroup[102X ([14X46.3-1[114X). For more information on possible values of
  [3Xfilt[103X  see  section  ([14X50.1[114X).  Methods  are also available for permutation and
  matrix   groups   (of  minimal  degree  and  minimal  dimension  in  coprime
  characteristic).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDicyclicGroup(24);[127X[104X
    [4X[28X<pc group of size 24 with 4 generators>[128X[104X
    [4X[25Xgap>[125X [27Xg:=QuaternionGroup(IsMatrixGroup,CF(16),32);[127X[104X
    [4X[28XGroup([ [ [ 0, 1 ], [ -1, 0 ] ], [ [ E(16), 0 ], [ 0, -E(16)^7 ] ] ])[128X[104X
  [4X[32X[104X
  
  [1X50.1-9 IsGeneralisedQuaternionGroup[101X
  
  [33X[1;0Y[29X[2XIsGeneralisedQuaternionGroup[102X( [3XG[103X ) [32X property[133X
  [33X[1;0Y[29X[2XIsQuaternionGroup[102X( [3XG[103X ) [32X property[133X
  [33X[1;0Y[29X[2XGeneralisedQuaternionGenerators[102X( [3XG[103X ) [32X attribute[133X
  [33X[1;0Y[29X[2XQuaternionGenerators[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0Y[2XIsGeneralisedQuaternionGroup[102X  indicates whether the group [3XG[103X is a generalized
  quaternion  group of size [22XN = 2^(k+1)[122X, [22Xk >= 2[122X. If it is, methods may set the
  attribute  [2XGeneralisedQuaternionGenerators[102X  to  [[3Xt[103X,[3Xs[103X], where [3Xt[103X and [3Xs[103X are two
  elements  such that [3XG[103X = [22X⟨ t, s | s^(2^k) = 1, t^2 = s^(2^k-1), s^t = s^-1 ⟩[122X.
  [2XIsQuaternionGroup[102X   and  [2XQuaternionGenerators[102X  are  provided  for  backwards
  compatibility with existing code.[133X
  
  [1X50.1-10 ExtraspecialGroup[101X
  
  [33X[1;0Y[29X[2XExtraspecialGroup[102X( [[3Xfilt[103X, ][3Xorder[103X, [3Xexp[103X ) [32X function[133X
  
  [33X[0;0YLet  [3Xorder[103X  be  of  the  form [22Xp^{2n+1}[122X, for a prime integer [22Xp[122X and a positive
  integer  [22Xn[122X.  [2XExtraspecialGroup[102X returns the extraspecial group of order [3Xorder[103X
  that is determined by [3Xexp[103X, in the category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [22Xp[122X  is  odd  then  admissible values of [3Xexp[103X are the exponent of the group
  (either  [22Xp[122X  or  [22Xp^2[122X) or one of [10X'+'[110X, [10X"+"[110X, [10X'-'[110X, [10X"-"[110X. For [22Xp = 2[122X, only the above
  plus or minus signs are admissible.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsPcGroup[102X ([14X46.3-1[114X). For more information
  on possible values of [3Xfilt[103X see section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XExtraspecialGroup( 27, 3 );[127X[104X
    [4X[28X<pc group of size 27 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XExtraspecialGroup( 27, '+' );[127X[104X
    [4X[28X<pc group of size 27 with 3 generators>[128X[104X
    [4X[25Xgap>[125X [27XExtraspecialGroup( 8, "-" );[127X[104X
    [4X[28X<pc group of size 8 with 3 generators>[128X[104X
  [4X[32X[104X
  
  
  [1X50.1-11 [33X[0;0YAlternatingGroup[133X[101X
  
  [33X[1;0Y[29X[2XAlternatingGroup[102X( [[3Xfilt[103X, ][3Xdeg[103X ) [32X function[133X
  [33X[1;0Y[29X[2XAlternatingGroup[102X( [[3Xfilt[103X, ][3Xdom[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  the alternating group of degree [3Xdeg[103X in the category given by the
  filter  [3Xfilt[103X.  If [3Xfilt[103X is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X). For
  more  information  on  possible  values  of  [3Xfilt[103X see section ([14X50.1[114X). In the
  second  version, the function constructs the alternating group on the points
  given in the set [3Xdom[103X which must be a set of positive integers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAlternatingGroup(5);[127X[104X
    [4X[28XAlt( [ 1 .. 5 ] )[128X[104X
  [4X[32X[104X
  
  
  [1X50.1-12 [33X[0;0YSymmetricGroup[133X[101X
  
  [33X[1;0Y[29X[2XSymmetricGroup[102X( [[3Xfilt[103X, ][3Xdeg[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSymmetricGroup[102X( [[3Xfilt[103X, ][3Xdom[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  the  symmetric  group of degree [3Xdeg[103X in the category given by the
  filter  [3Xfilt[103X.  If [3Xfilt[103X is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X). For
  more  information  on  possible  values  of  [3Xfilt[103X see section ([14X50.1[114X). In the
  second  version,  the  function constructs the symmetric group on the points
  given in the set [3Xdom[103X which must be a set of positive integers.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSymmetricGroup(10);[127X[104X
    [4X[28XSym( [ 1 .. 10 ] )[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  permutation  groups  provide  special treatment of symmetric and
  alternating groups, see [14X43.4[114X.[133X
  
  [1X50.1-13 MathieuGroup[101X
  
  [33X[1;0Y[29X[2XMathieuGroup[102X( [[3Xfilt[103X, ][3Xdegree[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  the  Mathieu group of degree [3Xdegree[103X in the category given by the
  filter [3Xfilt[103X, where [3Xdegree[103X must be in the set [22X{ 9, 10, 11, 12, 21, 22, 23, 24
  }[122X.  If  [3Xfilt[103X  is  not  given  it  defaults to [2XIsPermGroup[102X ([14X43.1-1[114X). For more
  information on possible values of [3Xfilt[103X see section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XMathieuGroup( 11 );[127X[104X
    [4X[28XGroup([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ])[128X[104X
  [4X[32X[104X
  
  [1X50.1-14 SuzukiGroup[101X
  
  [33X[1;0Y[29X[2XSuzukiGroup[102X( [[3Xfilt[103X, ][3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSz[102X( [[3Xfilt[103X, ][3Xq[103X ) [32X function[133X
  
  [33X[0;0YConstructs  a  group  isomorphic  to the Suzuki group Sz( [3Xq[103X ) over the field
  with [3Xq[103X elements, where [3Xq[103X is a non-square power of [22X2[122X.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group is the Suzuki group itself. For more information on possible values of
  [3Xfilt[103X see section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSuzukiGroup( 32 );[127X[104X
    [4X[28XSz(32)[128X[104X
  [4X[32X[104X
  
  [1X50.1-15 ReeGroup[101X
  
  [33X[1;0Y[29X[2XReeGroup[102X( [[3Xfilt[103X, ][3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XRee[102X( [[3Xfilt[103X, ][3Xq[103X ) [32X function[133X
  
  [33X[0;0YConstructs  a  group isomorphic to the Ree group [22X^2G_2(q)[122X where [22Xq = 3^{1+2m}[122X
  for [22Xm[122X a non-negative integer.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is  not  given  it  defaults  to  [2XIsMatrixGroup[102X  ([14X44.1-1[114X)  and the
  generating  matrices  are  based  on [KLM01].  (No  particular  choice  of a
  generating  set  is  guaranteed.) For more information on possible values of
  [3Xfilt[103X see section ([14X50.1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XReeGroup( 27 );[127X[104X
    [4X[28XRee(27)[128X[104X
  [4X[32X[104X
  
  
  [1X50.1-16 [33X[0;0YGenerator Names[133X[101X
  
  [33X[0;0YFor  groups  created  as  finitely  presented  groups,  including polycyclic
  groups, the generators are labelled, by default, with a letter and a number.
  It  is possible to influence this naming with the option [10XgeneratorNames[110X, see
  Section [14X4.12-2[114X. If this option holds a string, then the generators are named
  with  this  string  and  sequential  numbers starting with [10X1[110X. If this option
  holds  a  list of sufficient length consisting of nonempty strings, then the
  generator names are taken from this list, in order.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(AbelianGroup([5,7]));[127X[104X
    [4X[28X[ f1, f2 ][128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(AbelianGroup([5,7]:generatorNames:="a"));[127X[104X
    [4X[28X[ a1, a2 ][128X[104X
    [4X[25Xgap>[125X [27XGeneratorsOfGroup(AbelianGroup([5,7]:generatorNames:=["u","v","w"]));[127X[104X
    [4X[28X[ u, v ][128X[104X
    [4X[25Xgap>[125X [27XAsSSortedList(DihedralGroup(12:generatorNames:="a"));[127X[104X
    [4X[28X[ <identity> of ..., a1, a2, a3, a1*a2, a1*a3, a2*a3, a3^2, a1*a2*a3,[128X[104X
    [4X[28X  a1*a3^2, a2*a3^2, a1*a2*a3^2 ][128X[104X
    [4X[25Xgap>[125X [27XAsSSortedList(DihedralGroup(12:generatorNames:=["a","b","c"]));[127X[104X
    [4X[28X[ <identity> of ..., a, b, c, a*b, a*c, b*c, c^2, a*b*c, a*c^2, b*c^2,[128X[104X
    [4X[28X  a*b*c^2 ][128X[104X
  [4X[32X[104X
  
  
  [1X50.2 [33X[0;0YClassical Groups[133X[101X
  
  [33X[0;0YThe following functions return classical groups.[133X
  
  [33X[0;0YFor  the  linear, symplectic, and unitary groups (the latter in dimension at
  least  [22X3[122X),  the generators are taken from [Tay87]. For the unitary groups in
  dimension  [22X2[122X,  the  isomorphism  of  SU[22X(2,q)[122X  and  SL[22X(2,q)[122X  is used, see for
  example [Hup67].[133X
  
  [33X[0;0YThe  generators  of  the  general  and  special  orthogonal groups are taken
  from [IE94]  and  [KL90],  except  that  the generators of the groups in odd
  dimension  in  even  characteristic are constructed via the isomorphism to a
  symplectic group, see for example [Car72].[133X
  
  [33X[0;0YThe generators of the groups [22XΩ^ϵ(d, q)[122X are taken from [RT98], except that in
  odd  dimension and even characteristic, the generators of SO[22X(d, q)[122X are taken
  for [22XΩ(d, q)[122X. Note that the generators claimed in [RT98, Section 4.5 and 4.6]
  do  not  describe  orthogonal  groups,  one  would  have  to transpose these
  matrices  in  order  to  get  groups  that  respect  the required forms. The
  matrices  from [RT98] generate groups of the right isomorphism types but not
  orthogonal  groups,  except  in  the  case [22X(d,q) = (5,2)[122X, where the matrices
  from [RT98] generate the simple group [22XS_4(2)'[122X and not the group [22XS_4(2)[122X.[133X
  
  [33X[0;0YThe generators for the semilinear groups are constructed from the generators
  of  the  corresponding  linear  groups  plus  one  additional generator that
  describes the action of the group of field automorphisms; for prime integers
  [22Xp[122X  and positive integers [22Xf[122X, this yields the matrix groups [22XGamma[122XL[22X(d, p^f)[122X and
  [22XSigma[122XL[22X(d,  p^f)[122X  as  groups  of  [22Xd  f  ×  df[122X  matrices over the field with [22Xp[122X
  elements.[133X
  
  [33X[0;0YFor  symplectic  and  orthogonal  matrix  groups  returned  by the functions
  described  below,  the invariant bilinear form is stored as the value of the
  attribute   [2XInvariantBilinearForm[102X   ([14X44.5-1[114X).   Analogously,  the  invariant
  sesquilinear  form defining the unitary groups is stored as the value of the
  attribute  [2XInvariantSesquilinearForm[102X  ([14X44.5-3[114X)). The defining quadratic form
  of   orthogonal   groups   is   stored   as   the  value  of  the  attribute
  [2XInvariantQuadraticForm[102X ([14X44.5-5[114X).[133X
  
  [33X[0;0YNote  that  due  to  the different sources for the generators, the invariant
  forms  for  the  groups [22XΩ(e,d,q)[122X are in general different from the forms for
  SO[22X(e,d,q)[122X  and  GO[22X(e,d,q)[122X. If version at least 1.2.6 of the [5XForms[105X package is
  loaded then compatible groups can be created by specifying the desired form,
  see the sections below.[133X
  
  
  [1X50.2-1 [33X[0;0YGeneralLinearGroup[133X[101X
  
  [33X[1;0Y[29X[2XGeneralLinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3XR[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGL[102X( [[3Xfilt[103X, ][3Xd[103X, [3XR[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGeneralLinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGL[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0YThe first two forms construct a group isomorphic to the general linear group
  GL( [3Xd[103X, [3XR[103X ) of all [22X[3Xd[103X × [3Xd[103X[122X matrices that are invertible over the ring [3XR[103X, in the
  category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YThe  third  and  the fourth form construct the general linear group over the
  finite field with [3Xq[103X elements.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group  is  the  general linear group as a matrix group in its natural action
  (see also [2XIsNaturalGL[102X ([14X44.4-2[114X), [2XIsNaturalGLnZ[102X ([14X44.6-4[114X)).[133X
  
  [33X[0;0YCurrently  supported  rings [3XR[103X are finite fields, the ring [2XIntegers[102X ([14X14[114X), and
  residue class rings [10XIntegers mod [3Xm[103X[10X[110X, see [14X14.5[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGL(4,3);[127X[104X
    [4X[28XGL(4,3)[128X[104X
    [4X[25Xgap>[125X [27XGL(2,Integers);[127X[104X
    [4X[28XGL(2,Integers)[128X[104X
    [4X[25Xgap>[125X [27XGL(3,Integers mod 12);[127X[104X
    [4X[28XGL(3,Z/12Z)[128X[104X
  [4X[32X[104X
  
  [33X[0;0YUsing  the  [2XOnLines[102X  ([14X41.2-12[114X)  operation  it  is  possible  to  obtain  the
  corresponding projective groups in a permutation action:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:=GL(4,3);;Size(g);[127X[104X
    [4X[28X24261120[128X[104X
    [4X[25Xgap>[125X [27Xpgl:=Action(g,Orbit(g,Z(3)^0*[1,0,0,0],OnLines),OnLines);;[127X[104X
    [4X[25Xgap>[125X [27XSize(pgl);[127X[104X
    [4X[28X12130560[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  you  are  interested only in the projective group as a permutation group
  and not in the correspondence between its moved points and the points in the
  projective space, you can also use [2XPGL[102X ([14X50.2-11[114X).[133X
  
  
  [1X50.2-2 [33X[0;0YSpecialLinearGroup[133X[101X
  
  [33X[1;0Y[29X[2XSpecialLinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3XR[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSL[102X( [[3Xfilt[103X, ][3Xd[103X, [3XR[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSpecialLinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSL[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0YThe first two forms construct a group isomorphic to the special linear group
  SL(  [3Xd[103X, [3XR[103X ) of all those [22X[3Xd[103X × [3Xd[103X[122X matrices over the ring [3XR[103X whose determinant is
  the identity of [3XR[103X, in the category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YThe  third  and  the fourth form construct the special linear group over the
  finite field with [3Xq[103X elements.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group  is  the  special linear group as a matrix group in its natural action
  (see also [2XIsNaturalSL[102X ([14X44.4-4[114X), [2XIsNaturalSLnZ[102X ([14X44.6-5[114X)).[133X
  
  [33X[0;0YCurrently  supported  rings [3XR[103X are finite fields, the ring [2XIntegers[102X ([14X14[114X), and
  residue class rings [10XIntegers mod [3Xm[103X[10X[110X, see [14X14.5[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSpecialLinearGroup(2,2);[127X[104X
    [4X[28XSL(2,2)[128X[104X
    [4X[25Xgap>[125X [27XSL(3,Integers);[127X[104X
    [4X[28XSL(3,Integers)[128X[104X
    [4X[25Xgap>[125X [27XSL(4,Integers mod 4);[127X[104X
    [4X[28XSL(4,Z/4Z)[128X[104X
  [4X[32X[104X
  
  [1X50.2-3 GeneralUnitaryGroup[101X
  
  [33X[1;0Y[29X[2XGeneralUnitaryGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XGeneralUnitaryGroup[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGU[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XGU[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a  group  isomorphic  to the general unitary group GU( [3Xd[103X, [3Xq[103X ) of
  those  [22X[3Xd[103X  × [3Xd[103X[122X matrices over the field with [22X[3Xq[103X^2[122X elements that respect a fixed
  nondegenerate sesquilinear form, in the category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group is the general unitary group itself.[133X
  
  [33X[0;0YIf  version  at  least 1.2.6 of the [5XForms[105X package is loaded then the desired
  sesquilinear form can be specified via [3Xform[103X, which can be either a matrix or
  a  form  object  in [2XIsHermitianForm[102X ([14XForms: IsHermitianForm[114X) or a group with
  stored  [2XInvariantSesquilinearForm[102X  ([14X44.5-3[114X)  value  (and  then  this form is
  taken).[133X
  
  [33X[0;0YA  given  [3Xform[103X determines [3Xd[103X, and also [3Xq[103X except if [3Xform[103X is a matrix that does
  not  store  its  [2XBaseDomain[102X ([14X26.3-1[114X) value. These parameters can be entered,
  and an error is signalled if they do not fit to the given [3Xform[103X.[133X
  
  [33X[0;0YIf  [3Xform[103X  is  not  given  then  a  default  is  chosen  as  described in the
  introduction to Section [14X50.2[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGeneralUnitaryGroup( 3, 5 );[127X[104X
    [4X[28XGU(3,5)[128X[104X
    [4X[25Xgap>[125X [27XGeneralUnitaryGroup( IsPermGroup, 3, 5 );[127X[104X
    [4X[28XPerm_GU(3,5)[128X[104X
  [4X[32X[104X
  
  [1X50.2-4 SpecialUnitaryGroup[101X
  
  [33X[1;0Y[29X[2XSpecialUnitaryGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSpecialUnitaryGroup[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSU[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSU[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a  group  isomorphic  to the special unitary group SU( [3Xd[103X, [3Xq[103X ) of
  those  [22X[3Xd[103X  × [3Xd[103X[122X matrices over the field with [22X[3Xq[103X^2[122X elements whose determinant is
  the   identity   of  the  field  and  that  respect  a  fixed  nondegenerate
  sesquilinear form, in the category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group is the special unitary group itself.[133X
  
  [33X[0;0YIf  version  at  least 1.2.6 of the [5XForms[105X package is loaded then the desired
  sesquilinear form can be specified via [3Xform[103X, which can be either a matrix or
  a  form  object  in [2XIsHermitianForm[102X ([14XForms: IsHermitianForm[114X) or a group with
  stored  [2XInvariantSesquilinearForm[102X  ([14X44.5-3[114X)  value  (and  then  this form is
  taken).[133X
  
  [33X[0;0YA  given  [3Xform[103X determines [3Xd[103X, and also [3Xq[103X except if [3Xform[103X is a matrix that does
  not  store  its  [2XBaseDomain[102X ([14X26.3-1[114X) value. These parameters can be entered,
  and an error is signalled if they do not fit to the given [3Xform[103X.[133X
  
  [33X[0;0YIf  [3Xform[103X  is  not  given  then  a  default  is  chosen  as  described in the
  introduction to Section [14X50.2[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSpecialUnitaryGroup( 3, 5 );[127X[104X
    [4X[28XSU(3,5)[128X[104X
    [4X[25Xgap>[125X [27XSpecialUnitaryGroup( IsPermGroup, 3, 5 );[127X[104X
    [4X[28XPerm_SU(3,5)[128X[104X
  [4X[32X[104X
  
  
  [1X50.2-5 [33X[0;0YSymplecticGroup[133X[101X
  
  [33X[1;0Y[29X[2XSymplecticGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSymplecticGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xring[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSymplecticGroup[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSp[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSp[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xring[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSp[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSP[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSP[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xring[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSP[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a group isomorphic to the symplectic group Sp( [3Xd[103X, [3Xq[103X ) of those [22X[3Xd[103X
  ×  [3Xd[103X[122X  matrices  over  the field with [3Xq[103X elements (respectively the ring [3Xring[103X)
  that respect a fixed nondegenerate symplectic form, in the category given by
  the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group is the symplectic group itself.[133X
  
  [33X[0;0YAt the moment finite fields or residue class rings [10XIntegers mod [3Xq[103X[10X[110X, with [3Xq[103X an
  odd prime power, are supported.[133X
  
  [33X[0;0YIf  version  at least 1.2.6 of the [5XForms[105X package is loaded and the arguments
  describe  a  matrix group over a finite field then the desired bilinear form
  can  be specified via [3Xform[103X, which can be either a matrix or a form object in
  [2XIsBilinearForm[102X    ([14XForms:   IsBilinearForm[114X)   or   a   group   with   stored
  [2XInvariantBilinearForm[102X ([14X44.5-1[114X) value (and then this form is taken).[133X
  
  [33X[0;0YA  given  [3Xform[103X  determines and [3Xd[103X, and also [3Xq[103X except if [3Xform[103X is a matrix that
  does  not  store  its  [2XBaseDomain[102X  ([14X26.3-1[114X)  value.  These parameters can be
  entered, and an error is signalled if they do not fit to the given [3Xform[103X.[133X
  
  [33X[0;0YIf  [3Xform[103X  is  not  given  then  a  default  is  chosen  as  described in the
  introduction to Section [14X50.2[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSymplecticGroup( 4, 2 );[127X[104X
    [4X[28XSp(4,2)[128X[104X
    [4X[25Xgap>[125X [27Xg:=SymplecticGroup(6,Integers mod 9);[127X[104X
    [4X[28XSp(6,Z/9Z)[128X[104X
    [4X[25Xgap>[125X [27XSize(g);[127X[104X
    [4X[28X95928796265538862080[128X[104X
    [4X[25Xgap>[125X [27XSymplecticGroup( IsPermGroup, 4, 2 );[127X[104X
    [4X[28XPerm_Sp(4,2)[128X[104X
  [4X[32X[104X
  
  [1X50.2-6 GeneralOrthogonalGroup[101X
  
  [33X[1;0Y[29X[2XGeneralOrthogonalGroup[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XGeneralOrthogonalGroup[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGO[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XGO[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a group isomorphic to the general orthogonal group GO( [3Xe[103X, [3Xd[103X, [3Xq[103X )
  of  those  [22X[3Xd[103X  ×  [3Xd[103X[122X  matrices  over  the field with [3Xq[103X elements that respect a
  non-singular  quadratic form (see [2XInvariantQuadraticForm[102X ([14X44.5-5[114X)) specified
  by [3Xe[103X, in the category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YThe  value  of  [3Xe[103X must be [22X0[122X for odd [3Xd[103X (and can optionally be omitted in this
  case),  respectively  one  of  [22X1[122X  or  [22X-1[122X for even [3Xd[103X. If [3Xfilt[103X is not given it
  defaults  to  [2XIsMatrixGroup[102X  ([14X44.1-1[114X), and the returned group is the general
  orthogonal group itself.[133X
  
  [33X[0;0YIf  version  at  least 1.2.6 of the [5XForms[105X package is loaded then the desired
  quadratic  form can be specified via [3Xform[103X, which can be either a matrix or a
  form  object  in  [2XIsQuadraticForm[102X  ([14XForms:  IsQuadraticForm[114X) or a group with
  stored [2XInvariantQuadraticForm[102X ([14X44.5-5[114X) value (and then this form is taken).[133X
  
  [33X[0;0YA  given [3Xform[103X determines [3Xe[103X and [3Xd[103X, and also [3Xq[103X except if [3Xform[103X is a matrix that
  does  not  store  its  [2XBaseDomain[102X  ([14X26.3-1[114X)  value.  These parameters can be
  entered, and an error is signalled if they do not fit to the given [3Xform[103X.[133X
  
  [33X[0;0YIf  [3Xform[103X  is  not  given  then  a  default  is  chosen  as  described in the
  introduction to Section [14X50.2[114X.[133X
  
  [33X[0;0YNote  that  in [KL90],  GO  is  defined  as the stabilizer [22X∆(V, F, κ)[122X of the
  quadratic form, up to scalars, whereas our GO is called [22XI(V, F, κ)[122X there.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XGeneralOrthogonalGroup( 5, 3 );[127X[104X
    [4X[28XGO(0,5,3)[128X[104X
    [4X[25Xgap>[125X [27XGeneralOrthogonalGroup( -1, 8, 2 );[127X[104X
    [4X[28XGO(-1,8,2)[128X[104X
    [4X[25Xgap>[125X [27XGeneralOrthogonalGroup( IsPermGroup, -1, 8, 2 );[127X[104X
    [4X[28XPerm_GO(-1,8,2)[128X[104X
  [4X[32X[104X
  
  [1X50.2-7 SpecialOrthogonalGroup[101X
  
  [33X[1;0Y[29X[2XSpecialOrthogonalGroup[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSpecialOrthogonalGroup[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSO[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XSO[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X function[133X
  
  [33X[0;0Yconstructs a group isomorphic to the special orthogonal group SO( [3Xe[103X, [3Xd[103X, [3Xq[103X ),
  which  is the subgroup of all those matrices in the general orthogonal group
  (see [2XGeneralOrthogonalGroup[102X  ([14X50.2-6[114X))  that  have  determinant  one, in the
  category  given by the filter [3Xfilt[103X. (The index of SO( [3Xe[103X, [3Xd[103X, [3Xq[103X ) in GO( [3Xe[103X, [3Xd[103X,
  [3Xq[103X  )  is  [22X2[122X  if [3Xq[103X is odd, and [22X1[122X if [3Xq[103X is even.) Also interesting is the group
  Omega(  [3Xe[103X, [3Xd[103X, [3Xq[103X ), see [2XOmega[102X ([14X50.2-8[114X), which is of index [22X2[122X in SO( [3Xe[103X, [3Xd[103X, [3Xq[103X ),
  except in the case [22X[3Xd[103X = 1[122X.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group is the special orthogonal group itself.[133X
  
  [33X[0;0YIf  version  at  least 1.2.6 of the [5XForms[105X package is loaded then the desired
  quadratic  form can be specified via [3Xform[103X, which can be either a matrix or a
  form  object  in  [2XIsQuadraticForm[102X  ([14XForms:  IsQuadraticForm[114X) or a group with
  stored [2XInvariantQuadraticForm[102X ([14X44.5-5[114X) value (and then this form is taken).[133X
  
  [33X[0;0YA  given [3Xform[103X determines [3Xe[103X and [3Xd[103X, and also [3Xq[103X except if [3Xform[103X is a matrix that
  does  not  store  its  [2XBaseDomain[102X  ([14X26.3-1[114X)  value.  These parameters can be
  entered, and an error is signalled if they do not fit to the given [3Xform[103X.[133X
  
  [33X[0;0YIf  [3Xform[103X  is  not  given  then  a  default  is  chosen  as  described in the
  introduction to Section [14X50.2[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSpecialOrthogonalGroup( 5, 3 );[127X[104X
    [4X[28XSO(0,5,3)[128X[104X
    [4X[25Xgap>[125X [27XSpecialOrthogonalGroup( -1, 8, 2 );  # here SO and GO coincide[127X[104X
    [4X[28XGO(-1,8,2)[128X[104X
    [4X[25Xgap>[125X [27XSpecialOrthogonalGroup( IsPermGroup, 5, 3 );[127X[104X
    [4X[28XPerm_SO(0,5,3)[128X[104X
  [4X[32X[104X
  
  [1X50.2-8 Omega[101X
  
  [33X[1;0Y[29X[2XOmega[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X[, [3Xform[103X] ) [32X operation[133X
  [33X[1;0Y[29X[2XOmega[102X( [[3Xfilt[103X, ][3Xform[103X ) [32X operation[133X
  
  [33X[0;0Yconstructs  a  group  isomorphic  to  the  group [22XΩ[122X( [3Xe[103X, [3Xd[103X, [3Xq[103X ) of those [22X[3Xd[103X × [3Xd[103X[122X
  matrices  over  the  field  with  [3Xq[103X  elements  that  respect  a non-singular
  quadratic  form  (see [2XInvariantQuadraticForm[102X  ([14X44.5-5[114X))  specified by [3Xe[103X, and
  that have square spinor norm in odd characteristic or Dickson invariant [22X0[122X in
  even characteristic, respectively, in the category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YFor  odd  [3Xq[103X  and [22X[3Xd[103X ≥ 2[122X, this group has always index two in the corresponding
  special  orthogonal  group,  which will be conjugate in [22XGL(d,q)[122X to the group
  returned  by SO( [3Xe[103X, [3Xd[103X, [3Xq[103X ), see [2XSpecialOrthogonalGroup[102X ([14X50.2-7[114X), but may fix
  a different form (see [14X50.2[114X).[133X
  
  [33X[0;0YThe  value  of  [3Xe[103X must be [22X0[122X for odd [3Xd[103X (and can optionally be omitted in this
  case),  respectively  one  of  [22X1[122X  or  [22X-1[122X for even [3Xd[103X. If [3Xfilt[103X is not given it
  defaults  to  [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned group is the group [22XΩ[122X(
  [3Xe[103X, [3Xd[103X, [3Xq[103X ) itself.[133X
  
  [33X[0;0YIf  version  at  least 1.2.6 of the [5XForms[105X package is loaded then the desired
  quadratic  form can be specified via [3Xform[103X, which can be either a matrix or a
  form  object  in  [2XIsQuadraticForm[102X  ([14XForms:  IsQuadraticForm[114X) or a group with
  stored [2XInvariantQuadraticForm[102X ([14X44.5-5[114X) value (and then this form is taken).[133X
  
  [33X[0;0YA  given [3Xform[103X determines [3Xe[103X and [3Xd[103X, and also [3Xq[103X except if [3Xform[103X is a matrix that
  does  not  store  its  [2XBaseDomain[102X  ([14X26.3-1[114X)  value.  These parameters can be
  entered, and an error is signalled if they do not fit to the given [3Xform[103X.[133X
  
  [33X[0;0YIf  [3Xform[103X  is  not  given  then  a  default  is  chosen  as  described in the
  introduction to Section [14X50.2[114X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg:= Omega( 3, 5 );  StructureDescription( g );[127X[104X
    [4X[28XOmega(0,3,5)[128X[104X
    [4X[28X"A5"[128X[104X
    [4X[25Xgap>[125X [27Xg:= Omega( 1, 4, 4 );  StructureDescription( g );[127X[104X
    [4X[28XOmega(+1,4,4)[128X[104X
    [4X[28X"A5 x A5"[128X[104X
    [4X[25Xgap>[125X [27Xg:= Omega( -1, 4, 3 );  StructureDescription( g );[127X[104X
    [4X[28XOmega(-1,4,3)[128X[104X
    [4X[28X"A6"[128X[104X
    [4X[25Xgap>[125X [27Xg:= Omega( IsPermGroup, 1, 6, 2 );  StructureDescription( g );[127X[104X
    [4X[28XPerm_Omega(+1,6,2)[128X[104X
    [4X[28X"A8"[128X[104X
    [4X[25Xgap>[125X [27XIsSubset( GO( 3, 5 ), Omega( 3, 5 ) );  # different forms![127X[104X
    [4X[28Xfalse[128X[104X
  [4X[32X[104X
  
  [1X50.2-9 GeneralSemilinearGroup[101X
  
  [33X[1;0Y[29X[2XGeneralSemilinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XGammaL[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Y[2XGeneralSemilinearGroup[102X  returns a group isomorphic to the general semilinear
  group [22XΓ[122XL( [3Xd[103X, [3Xq[103X ) of semilinear mappings of the vector space [10XGF( [110X[3Xq[103X[10X )^[110X[3Xd[103X.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group  consists of matrices of dimension [3Xd[103X [22Xf[122X over the field with [22Xp[122X elements,
  where [3Xq[103X [22X= p^f[122X, for a prime integer [22Xp[122X.[133X
  
  [1X50.2-10 SpecialSemilinearGroup[101X
  
  [33X[1;0Y[29X[2XSpecialSemilinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XSigmaL[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Y[2XSpecialSemilinearGroup[102X  returns a group isomorphic to the special semilinear
  group  [22XΣ[122XL( [3Xd[103X, [3Xq[103X ) of those semilinear mappings of the vector space [10XGF( [110X[3Xq[103X[10X )^[110X[3Xd[103X
  (see [2XGeneralSemilinearGroup[102X ([14X50.2-9[114X)) whose linear part has determinant one.[133X
  
  [33X[0;0YIf [3Xfilt[103X is not given it defaults to [2XIsMatrixGroup[102X ([14X44.1-1[114X), and the returned
  group  consists of matrices of dimension [3Xd[103X [22Xf[122X over the field with [22Xp[122X elements,
  where [3Xq[103X [22X= p^f[122X, for a prime integer [22Xp[122X.[133X
  
  [1X50.2-11 ProjectiveGeneralLinearGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveGeneralLinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPGL[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Yconstructs a group isomorphic to the projective general linear group PGL( [3Xd[103X,
  [3Xq[103X  )  of  those  [22X[3Xd[103X  ×  [3Xd[103X[122X matrices over the field with [3Xq[103X elements, modulo the
  centre, in the category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space.[133X
  
  [1X50.2-12 ProjectiveSpecialLinearGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveSpecialLinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPSL[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Yconstructs a group isomorphic to the projective special linear group PSL( [3Xd[103X,
  [3Xq[103X ) of those [22X[3Xd[103X × [3Xd[103X[122X matrices over the field with [3Xq[103X elements whose determinant
  is  the  identity  of the field, modulo the centre, in the category given by
  the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space.[133X
  
  [1X50.2-13 ProjectiveGeneralUnitaryGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveGeneralUnitaryGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPGU[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a  group isomorphic to the projective general unitary group PGU(
  [3Xd[103X, [3Xq[103X ) of those [22X[3Xd[103X × [3Xd[103X[122X matrices over the field with [22X[3Xq[103X^2[122X elements that respect
  a  fixed nondegenerate sesquilinear form, modulo the centre, in the category
  given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space.[133X
  
  [1X50.2-14 ProjectiveSpecialUnitaryGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveSpecialUnitaryGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPSU[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a  group isomorphic to the projective special unitary group PSU(
  [3Xd[103X, [3Xq[103X ) of those [22X[3Xd[103X × [3Xd[103X[122X matrices over the field with [22X[3Xq[103X^2[122X elements that respect
  a  fixed  nondegenerate sesquilinear form and have determinant 1, modulo the
  centre, in the category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space.[133X
  
  [1X50.2-15 ProjectiveSymplecticGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveSymplecticGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPSP[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPSp[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Yconstructs a group isomorphic to the projective symplectic group PSp([3Xd[103X,[3Xq[103X) of
  those  [22X[3Xd[103X  ×  [3Xd[103X[122X  matrices over the field with [3Xq[103X elements that respect a fixed
  nondegenerate  symplectic  form, modulo the centre, in the category given by
  the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space.[133X
  
  [1X50.2-16 ProjectiveGeneralOrthogonalGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveGeneralOrthogonalGroup[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPGO[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a group isomorphic to the projective group PGO( [3Xe[103X, [3Xd[103X, [3Xq[103X ) of GO(
  [3Xe[103X,  [3Xd[103X,  [3Xq[103X ), modulo the centre (see [2XGeneralOrthogonalGroup[102X ([14X50.2-6[114X)), in the
  category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space.[133X
  
  [1X50.2-17 ProjectiveSpecialOrthogonalGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveSpecialOrthogonalGroup[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPSO[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Yconstructs  a group isomorphic to the projective group PSO( [3Xe[103X, [3Xd[103X, [3Xq[103X ) of SO(
  [3Xe[103X,  [3Xd[103X,  [3Xq[103X ), modulo the centre (see [2XSpecialOrthogonalGroup[102X ([14X50.2-7[114X)), in the
  category given by the filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space.[133X
  
  [1X50.2-18 ProjectiveOmega[101X
  
  [33X[1;0Y[29X[2XProjectiveOmega[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPOmega[102X( [[3Xfilt[103X, ][[3Xe[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Yconstructs a group isomorphic to the projective group P[22XΩ[122X( [3Xe[103X, [3Xd[103X, [3Xq[103X ) of [22XΩ[122X( [3Xe[103X,
  [3Xd[103X, [3Xq[103X ), modulo the centre (see [2XOmega[102X ([14X50.2-8[114X)), in the category given by the
  filter [3Xfilt[103X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space.[133X
  
  [1X50.2-19 ProjectiveGeneralSemilinearGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveGeneralSemilinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPGammaL[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Y[2XProjectiveGeneralSemilinearGroup[102X  returns  a  group isomorphic to the factor
  group  of  the  general semilinear group [10XGammaL([110X [3Xd[103X, [3Xq[103X [10X)[110X modulo the center of
  its normal subgroup [10XGL([110X [3Xd[103X, [3Xq[103X [10X)[110X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space [10XGF([110X[3Xq[103X[10X)^[110X[3Xd[103X.[133X
  
  [1X50.2-20 ProjectiveSpecialSemilinearGroup[101X
  
  [33X[1;0Y[29X[2XProjectiveSpecialSemilinearGroup[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XPSigmaL[102X( [[3Xfilt[103X, ][3Xd[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Y[2XProjectiveSpecialSemilinearGroup[102X  returns  a  group isomorphic to the factor
  group  of  the  special semilinear group [10XSigmaL([110X [3Xd[103X, [3Xq[103X [10X)[110X modulo the center of
  its normal subgroup [10XSL([110X [3Xd[103X, [3Xq[103X [10X)[110X.[133X
  
  [33X[0;0YIf  [3Xfilt[103X  is not given it defaults to [2XIsPermGroup[102X ([14X43.1-1[114X), and the returned
  group is the action on lines of the underlying vector space [10XGF([110X[3Xq[103X[10X)^[110X[3Xd[103X.[133X
  
  
  [1X50.3 [33X[0;0YConjugacy Classes in Classical Groups[133X[101X
  
  [33X[0;0YFor  general  and  special  linear  groups  (see [2XGeneralLinearGroup[102X ([14X50.2-1[114X)
  and [2XSpecialLinearGroup[102X  ([14X50.2-2[114X))  [5XGAP[105X  has  an efficient method to generate
  representatives  of  the  conjugacy  classes.  This uses results from linear
  algebra  on  normal  forms of matrices. If you know how to do this for other
  types of classical groups, please, tell us.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xg := SL(4,9);[127X[104X
    [4X[28XSL(4,9)[128X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClasses(g);[127X[104X
    [4X[28X861[128X[104X
    [4X[25Xgap>[125X [27Xcl := ConjugacyClasses(g);;[127X[104X
    [4X[25Xgap>[125X [27XLength(cl);[127X[104X
    [4X[28X861[128X[104X
  [4X[32X[104X
  
  [1X50.3-1 NrConjugacyClassesGL[101X
  
  [33X[1;0Y[29X[2XNrConjugacyClassesGL[102X( [3Xn[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesGU[102X( [3Xn[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesSL[102X( [3Xn[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesSU[102X( [3Xn[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesPGL[102X( [3Xn[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesPGU[102X( [3Xn[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesPSL[102X( [3Xn[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesPSU[102X( [3Xn[103X, [3Xq[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesSLIsogeneous[102X( [3Xn[103X, [3Xq[103X, [3Xf[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrConjugacyClassesSUIsogeneous[102X( [3Xn[103X, [3Xq[103X, [3Xf[103X ) [32X function[133X
  
  [33X[0;0YThe  first of these functions compute for given positive integer [3Xn[103X and prime
  power  [3Xq[103X the number of conjugacy classes in the classical groups GL( [3Xn[103X, [3Xq[103X ),
  GU(  [3Xn[103X,  [3Xq[103X ), SL( [3Xn[103X, [3Xq[103X ), SU( [3Xn[103X, [3Xq[103X ), PGL( [3Xn[103X, [3Xq[103X ), PGU( [3Xn[103X, [3Xq[103X ), PSL( [3Xn[103X, [3Xq[103X ),
  PSL(  [3Xn[103X,  [3Xq[103X  ),  respectively.  (See  also  [2XConjugacyClasses[102X  ([14X39.10-2[114X)  and
  Section [14X50.2[114X.)[133X
  
  [33X[0;0YFor  each divisor [3Xf[103X of [3Xn[103X there is a group of Lie type with the same order as
  SL(  [3Xn[103X,  [3Xq[103X ), such that its derived subgroup modulo its center is isomorphic
  to  PSL(  [3Xn[103X,  [3Xq[103X  ).  The  various  such groups with fixed [3Xn[103X and [3Xq[103X are called
  [13Xisogeneous[113X.  (Depending on congruence conditions on [3Xq[103X and [3Xn[103X several of these
  groups      may      actually      be      isomorphic.)     The     function
  [2XNrConjugacyClassesSLIsogeneous[102X  computes  the number of conjugacy classes in
  this  group. The extreme cases [3Xf[103X [22X= 1[122X and [3Xf[103X [22X= n[122X lead to the groups SL( [3Xn[103X, [3Xq[103X )
  and PGL( [3Xn[103X, [3Xq[103X ), respectively.[133X
  
  [33X[0;0YThe  function  [2XNrConjugacyClassesSUIsogeneous[102X  is  the analogous one for the
  corresponding unitary groups.[133X
  
  [33X[0;0YThe formulae for the number of conjugacy classes are taken from [Mac81].[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClassesGL(24,27);[127X[104X
    [4X[28X22528399544939174406067288580609952[128X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClassesPSU(19,17);[127X[104X
    [4X[28X15052300411163848367708[128X[104X
    [4X[25Xgap>[125X [27XNrConjugacyClasses(SL(16,16));[127X[104X
    [4X[28X1229782938228219920[128X[104X
  [4X[32X[104X
  
  
  [1X50.4 [33X[0;0YConstructors for Basic Groups[133X[101X
  
  [33X[0;0YAll functions described in the previous sections call constructor operations
  to do the work. The names of the constructors are obtained from the names of
  the functions by appending [10X"Cons"[110X, so for example [2XCyclicGroup[102X ([14X50.1-2[114X) calls
  the constructor[133X
  
  [33X[0;0Y[10XCyclicGroupCons( [3Xcat[103X[10X, [3Xn[103X[10X )[110X[133X
  
  [33X[0;0YThe  first  argument  [3Xcat[103X  for  each  method of this constructor must be the
  category  for  which  the  method  is  installed. For example the method for
  constructing   a   cyclic   permutation   group   is  installed  as  follows
  (see [2XInstallMethod[102X ([14X78.3-1[114X) for the meaning of the arguments.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XInstallMethod( CyclicGroupCons,[128X[104X
    [4X[28X    "regular perm group",[128X[104X
    [4X[28X    true,[128X[104X
    [4X[28X    [ IsPermGroup and IsRegularProp and IsFinite, IsInt and IsPosRat ], 0,[128X[104X
    [4X[28X    function( filter, n )[128X[104X
    [4X[28X[128X[104X
    [4X[28X    ...[128X[104X
    [4X[28X[128X[104X
    [4X[28X    end );[128X[104X
  [4X[32X[104X
  
  
  [1X50.5 [33X[0;0YSelection Functions[133X[101X
  
  [33X[0;0Y[10XAll[3XLibrary[103X[10XGroups( [3Xfun1[103X[10X, [3Xval1[103X[10X, ... )[110X[133X
  
  [33X[0;0YFor  a  number  of the group libraries two [13Xselection functions[113X are provided.
  Each  [10XAll[3XLibrary[103X[10XGroups[110X  selection  function permits one to select [13Xall[113X groups
  from the library [3XLibrary[103X that have a given set of properties. Currently, the
  library  selection  functions  provided,  of  this  type, are [2XAllSmallGroups[102X
  ([14Xsmallgrp:     AllSmallGroups[114X),    [2XAllIrreducibleSolvableGroups[102X    ([14Xprimgrp:
  AllIrreducibleSolvableGroups[114X),         [2XAllTransitiveGroups[102X        ([14Xtransgrp:
  AllTransitiveGroups[114X),  and [2XAllPrimitiveGroups[102X ([14Xprimgrp: AllPrimitiveGroups[114X).
  Corresponding  to  each  of  these  there is a [10XOne[3XLibrary[103X[10XGroup[110X function (see
  below) which returns at most one group.[133X
  
  [33X[0;0YThese functions take an arbitrary number of pairs (but at least one pair) of
  arguments.  The  first  argument  in  such  a pair is a function that can be
  applied  to  the  groups in the library, and the second argument is either a
  single  value  that  this  function  must return in order to have this group
  included  in  the  selection,  or  a  list  of such values. For the function
  [2XAllSmallGroups[102X  ([14Xsmallgrp:  AllSmallGroups[114X)  the first such function must be
  [2XSize[102X  ([14X30.4-6[114X),  and,  unlike  the  other  library  selection  functions, it
  supports   an   alternative   syntax   where   [2XSize[102X   ([14X30.4-6[114X)   is  omitted
  (see [2XAllSmallGroups[102X      ([14Xsmallgrp:      AllSmallGroups[114X)).     Also,     see
  [2XAllIrreducibleSolvableGroups[102X  ([14Xprimgrp:  AllIrreducibleSolvableGroups[114X),  for
  details pertaining to this function.[133X
  
  [33X[0;0YFor  an  example,  let us consider the selection function for the library of
  transitive  groups (also see [14X'transgrp: Transitive Permutation Groups'[114X). The
  command[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XAllTransitiveGroups(NrMovedPoints,[10..15],[127X[104X
    [4X[25X>[125X [27X                       Size,         [1..100],[127X[104X
    [4X[25X>[125X [27X                       IsAbelian,    false    );[127X[104X
  [4X[32X[104X
  
  [33X[0;0Yreturns  a  list  of all transitive groups with degree between 10 and 15 and
  size less than 100 that are not abelian.[133X
  
  [33X[0;0YThus  [10XAllTransitiveGroups[110X  behaves  as  if  it was implemented by a function
  similar  to  the  one defined below, where [10XTransitiveGroupsList[110X is a list of
  all  transitive  groups.  (Note  that  in the definition below we assume for
  simplicity  that  [10XAllTransitiveGroups[110X  accepts exactly 4 arguments. It is of
  course  obvious  how  to  change  this definition so that the function would
  accept a variable number of arguments.)[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28XAllTransitiveGroups := function( fun1, val1, fun2, val2 )[128X[104X
    [4X[28Xlocal    groups, g, i;[128X[104X
    [4X[28X  groups := [];[128X[104X
    [4X[28X  for i  in [ 1 .. Length( TransitiveGroupsList ) ] do[128X[104X
    [4X[28X    g := TransitiveGroupsList[i];[128X[104X
    [4X[28X    if      fun1(g) = val1  or IsList(val1) and fun1(g) in val1[128X[104X
    [4X[28X        and fun2(g) = val2  or IsList(val2) and fun2(g) in val2[128X[104X
    [4X[28X     then[128X[104X
    [4X[28X      Add( groups, g );[128X[104X
    [4X[28X    fi;[128X[104X
    [4X[28X  od;[128X[104X
    [4X[28X  return groups;[128X[104X
    [4X[28Xend;[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the real selection functions are considerably more difficult, to
  improve  the  efficiency.  Most  important, each recognizes a certain set of
  properties  which  are precomputed for the library without having to compute
  them  anew  for  each  group. This will substantially speed up the selection
  process. In the description of each library we will list the properties that
  are stored for this library.[133X
  
  [33X[0;0Y[10XOne[3XLibrary[103X[10XGroup( [3Xfun1[103X[10X, [3Xval1[103X[10X, ... )[110X[133X
  
  [33X[0;0YFor  each  [10XAll[3XLibrary[103X[10XGroups[110X  function  (see  above) there is a corresponding
  function  [10XOne[3XLibrary[103X[10XGroup[110X  on  exactly  the  same arguments, i.e., there are
  [2XOneSmallGroup[102X    ([14Xsmallgrp:    OneSmallGroup[114X),   [2XOneIrreducibleSolvableGroup[102X
  ([14Xprimgrp:    OneIrreducibleSolvableGroup[114X),   [2XOneTransitiveGroup[102X   ([14Xtransgrp:
  OneTransitiveGroup[114X),  and  [2XOnePrimitiveGroup[102X  ([14Xprimgrp:  OnePrimitiveGroup[114X).
  Each  function  simply  returns  [13Xone[113X  group  in  the  library  that  has the
  prescribed  properties,  instead  of  [13Xall[113X such groups. It returns [9Xfail[109X if no
  such group exists in the library.[133X
  
  
  [1X50.6 [33X[0;0YFinite Perfect Groups[133X[101X
  
  [33X[0;0YThe [5XGAP[105X library of finite perfect groups provides, up to isomorphism, a list
  of  all  perfect  groups  whose  sizes  are less than [22X2⋅ 10^6[122X. The groups of
  orders  up to [22X10^6[122X have been enumerated by Derek F. Holt and Wilhelm Plesken
  and  published  in  their  book [21XPerfect Groups[121X [HP89]. For orders [22Xn = 86016[122X,
  368640,  or 737280 this work only counted the groups (but did not explicitly
  list  them), the groups of orders [22Xn = 61440[122X, 122880, 172032, 245760, 344064,
  491520, 688128, or 983040 were omitted.[133X
  
  [33X[0;0YWe  are  grateful  to Derek Holt and Wilhelm Plesken for making their groups
  available  to  the  [5XGAP[105X  community by contributing their files. It should be
  noted  that their book contains a lot of further information for many of the
  library  groups.  So  we  would  like to recommend it to any [5XGAP[105X user who is
  interested  in  the  groups.  The library of these has been brought into [5XGAP[105X
  format by Volkmar Felsch.[133X
  
  [33X[0;0YSeveral  additional  groups  omitted  from the book [21XPerfect Groups[121X have also
  been included. Two groups -- one of order 450000 with a factor group of type
  [22XA_6[122X  and  the one of order 962280 -- were found by Jack Schmidt in 2005. Two
  groups  of  order  243000 and one each of orders 729000, 871200, 878460 were
  found in 2020 by Alexander Hulpke.[133X
  
  [33X[0;0YThe  perfect  groups of size less than [22X2⋅ 10^6[122X which had not been classified
  in  the  work  of Holt and Plesken have been enumerated by Alexander Hulpke.
  They  are stored directly and provide less construction information in their
  names.[133X
  
  [33X[0;0YAs  all  groups are stored by presentations, a permutation representation is
  obtained  by  coset enumeration. Note that some of the library groups do not
  have  a faithful permutation representation of small degree. Computations in
  these groups may be rather time consuming.[133X
  
  [1X50.6-1 SizesPerfectGroups[101X
  
  [33X[1;0Y[29X[2XSizesPerfectGroups[102X(  ) [32X function[133X
  
  [33X[0;0YThis is the ordered list of all numbers up to [22X2⋅ 10^6[122X that occur as sizes of
  perfect  groups.  One  can  iterate  over part of the perfect groups library
  with:[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27Xfor n in Intersection([100..500],SizesPerfectGroups()) do[127X[104X
    [4X[25X>[125X [27X     for k in [1..NrPerfectGroups(n)] do[127X[104X
    [4X[25X>[125X [27X       pg := PerfectGroup(n,k);[127X[104X
    [4X[25X>[125X [27X     od;[127X[104X
    [4X[25X>[125X [27X   od;[127X[104X
  [4X[32X[104X
  
  
  [1X50.6-2 [33X[0;0YPerfectGroup[133X[101X
  
  [33X[1;0Y[29X[2XPerfectGroup[102X( [[3Xfilt[103X, ][3Xsize[103X[, [3Xn[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XPerfectGroup[102X( [[3Xfilt[103X, ][3Xsizenumberpair[103X ) [32X function[133X
  
  [33X[0;0Yreturns  a  group  which is isomorphic to the library group specified by the
  size  number  [10X[  [3Xsize[103X[10X,  [3Xn[103X[10X  ][110X  or  by  the two separate arguments [3Xsize[103X and [3Xn[103X,
  assuming  a  default  value of [22X[3Xn[103X = 1[122X. The optional argument [3Xfilt[103X defines the
  filter  in  which  the  group  is  returned.  Possible  filters  so  far are
  [2XIsPermGroup[102X ([14X43.1-1[114X) and [2XIsSubgroupFpGroup[102X ([14X47.1-1[114X). In the latter case, the
  generators  and  relators  used  coincide  with  those  given in [HP89]. The
  default filter is [2XIsPermGroup[102X ([14X43.1-1[114X).[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := PerfectGroup(IsPermGroup,6048,1);[127X[104X
    [4X[28XU3(3)[128X[104X
    [4X[25Xgap>[125X [27XG:=PerfectGroup(IsPermGroup,823080,2);[127X[104X
    [4X[28XA5 2^1 19^2 C 19^1[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints(G);[127X[104X
    [4X[28X6859[128X[104X
    [4X[25Xgap>[125X [27XG:=PerfectGroup(1866240,12);[127X[104X
    [4X[28XPG1866240.12[128X[104X
    [4X[25Xgap>[125X [27XNrMovedPoints(G);[127X[104X
    [4X[28X270[128X[104X
  [4X[32X[104X
  
  [1X50.6-3 PerfectIdentification[101X
  
  [33X[1;0Y[29X[2XPerfectIdentification[102X( [3XG[103X ) [32X attribute[133X
  
  [33X[0;0YThis  attribute  is  set  for  all  groups  obtained from the perfect groups
  library  and  has  the  value  [10X[[3Xsize[103X[10X,[3Xnr[103X[10X][110X if the group is obtained with these
  parameters from the library.[133X
  
  [1X50.6-4 NumberPerfectGroups[101X
  
  [33X[1;0Y[29X[2XNumberPerfectGroups[102X( [3Xsize[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrPerfectGroups[102X( [3Xsize[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNumberPerfectLibraryGroups[102X( [3Xsize[103X ) [32X function[133X
  [33X[1;0Y[29X[2XNrPerfectLibraryGroups[102X( [3Xsize[103X ) [32X function[133X
  
  [33X[0;0Yreturns  the  number  of non-isomorphic perfect groups of size [3Xsize[103X for each
  positive  integer [3Xsize[103X up to [22X2⋅10^6[122X. Additionally, for odd [3Xsize[103X an answer is
  returned  (odd  order  groups  are  solvable). For any other argument out of
  range it returns [9Xfail[109X. [3XNrPerfectGroups[103X is a synonym for [2XNumberPerfectGroups[102X.
  Moreover [3XNumberPerfectLibraryGroups[103X (and its synonym [3XNrPerfectLibraryGroups[103X)
  exist  for  historical  reasons,  and return 0 instead of fail for arguments
  outside the library scope.[133X
  
  [1X50.6-5 SizeNumbersPerfectGroups[101X
  
  [33X[1;0Y[29X[2XSizeNumbersPerfectGroups[102X( [3Xfactor1[103X, [3Xfactor2[103X, [3X...[103X ) [32X function[133X
  
  [33X[0;0Y[2XSizeNumbersPerfectGroups[102X returns a list of pairs, each entry consisting of a
  group  order and the number of those groups in the library of perfect groups
  that  contain  the  specified  factors  [3Xfactor1[103X,  [3Xfactor2[103X,  ...  among their
  composition factors.[133X
  
  [33X[0;0YEach  argument  must  either  be the name of a nonabelian simple group or an
  integer  which  stands  for  the  product of the sizes of one or more cyclic
  factors. (In fact, the function replaces all integers among the arguments by
  their product.)[133X
  
  [33X[0;0YThe following text strings are accepted as simple group names.[133X
  
  [30X    [33X[0;6Y[10XA[3Xn[103X[10X[110X  or  [10XA([3Xn[103X[10X)[110X for the alternating groups [22XA_[3Xn[103X[122X, [22X5 ≤ n ≤ 9[122X, for example [10XA5[110X
        or [10XA(6)[110X.[133X
  
  [30X    [33X[0;6Y[10XL[3Xn[103X[10X([3Xq[103X[10X)[110X  or [10XL([3Xn[103X[10X,[3Xq[103X[10X)[110X for PSL[22X(n,q)[122X, where [22Xn ∈ { 2, 3 }[122X and [22Xq[122X a prime power,
        ranging[133X
  
        [30X    [33X[0;12Yfor [22Xn = 2[122X from 4 to 125[133X
  
        [30X    [33X[0;12Yfor [22Xn = 3[122X from 2 to 5[133X
  
  [30X    [33X[0;6Y[10XU[3Xn[103X[10X([3Xq[103X[10X)[110X  or [10XU([3Xn[103X[10X,[3Xq[103X[10X)[110X for PSU[22X(n,q)[122X, where [22Xn ∈ { 3, 4 }[122X and [22Xq[122X a prime power,
        ranging[133X
  
        [30X    [33X[0;12Yfor [22Xn = 3[122X from 3 to 5[133X
  
        [30X    [33X[0;12Yfor [22Xn = 4[122X from 2 to 2[133X
  
  [30X    [33X[0;6Y[10XSp4(4)[110X or [10XS(4,4)[110X for the symplectic group Sp[22X(4,4)[122X,[133X
  
  [30X    [33X[0;6Y[10XSz(8)[110X for the Suzuki group Sz[22X(8)[122X,[133X
  
  [30X    [33X[0;6Y[10XM[3Xn[103X[10X[110X or [10XM([3Xn[103X[10X)[110X for the Mathieu groups [22XM_11[122X, [22XM_12[122X, and [22XM_22[122X, and[133X
  
  [30X    [33X[0;6Y[10XJ[3Xn[103X[10X[110X or [10XJ([3Xn[103X[10X)[110X for the Janko groups [22XJ_1[122X and [22XJ_2[122X.[133X
  
  [33X[0;0YNote  that,  for most of the groups, the preceding list offers two different
  names  in order to be consistent with the notation used in [HP89] as well as
  with  the notation used in the [2XDisplayCompositionSeries[102X ([14X39.17-6[114X) command of
  [5XGAP[105X.  However, as the names are compared as text strings, you are restricted
  to the above choice. Even expressions like [10XL2(2^5)[110X are not accepted.[133X
  
  [33X[0;0YAs  the use of the term PSU[22X(n,q)[122X is not unique in the literature, we mention
  that  in  this library it denotes the factor group of SU[22X(n,q)[122X by its centre,
  where  SU[22X(n,q)[122X  is  the  group of all [22Xn × n[122X unitary matrices with entries in
  [22XGF(q^2)[122X and determinant 1.[133X
  
  [33X[0;0YThe  purpose  of the function is to provide a simple way to formulate a loop
  over all library groups which contain certain composition factors.[133X
  
  
  [1X50.6-6 [33X[0;0YDisplayInformationPerfectGroups[133X[101X
  
  [33X[1;0Y[29X[2XDisplayInformationPerfectGroups[102X( [3Xsize[103X[, [3Xn[103X] ) [32X function[133X
  [33X[1;0Y[29X[2XDisplayInformationPerfectGroups[102X( [3Xsizenumberpair[103X ) [32X function[133X
  
  [33X[0;0Y[2XDisplayInformationPerfectGroups[102X  displays  some invariants of the [3Xn[103X-th group
  of order [3Xsize[103X from the perfect groups library.[133X
  
  [33X[0;0YIf  no  value  of [3Xn[103X has been specified, the invariants will be displayed for
  all groups of size [3Xsize[103X available in the library.[133X
  
  [33X[0;0YAlternatively,  also  a  list  of  length  two  may  be  entered as the only
  argument, with entries [3Xsize[103X and [3Xn[103X.[133X
  
  [33X[0;0YThe information provided for [22XG[122X includes the following items:[133X
  
  [30X    [33X[0;6Ya  headline  containing  the  size number [10X[ [3Xsize[103X[10X, [3Xn[103X[10X ][110X of [22XG[122X in the form
        [10X[3Xsize[103X[10X.[3Xn[103X[10X[110X  (the  suffix [10X.[3Xn[103X[10X[110X will be suppressed if, up to isomorphism, [22XG[122X is
        the only perfect group of order [3Xsize[103X),[133X
  
  [30X    [33X[0;6Ya  message if [22XG[122X is simple or quasisimple, i.e., if the factor group of
        [22XG[122X by its centre is simple,[133X
  
  [30X    [33X[0;6Ythe  [21Xdescription[121X  of  the  structure  of  [22XG[122X as it is given by Holt and
        Plesken in [HP89] (see below),[133X
  
  [30X    [33X[0;6Ythe size of the centre of [22XG[122X (suppressed, if [22XG[122X is simple),[133X
  
  [30X    [33X[0;6Ythe prime decomposition of the size of [22XG[122X,[133X
  
  [30X    [33X[0;6Yorbit  sizes  for  a faithful permutation representation of [22XG[122X which is
        provided by the library (see below),[133X
  
  [30X    [33X[0;6Ya  reference  to  each occurrence of [22XG[122X in the tables of section 5.3 of
        [HP89].  Each  of  these  references consists of a class number and an
        internal  number [22X(i,j)[122X under which [22XG[122X is listed in that class. For some
        groups,  there  is more than one reference because these groups belong
        to more than one of the classes in the book.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayInformationPerfectGroups( 30720, 3 );[127X[104X
    [4X[28X#I Perfect group 30720:  A5 ( 2^4 E N 2^1 E 2^4 ) A[128X[104X
    [4X[28X#I   size = 2^11*3*5  orbit size = 240[128X[104X
    [4X[28X#I   Holt-Plesken class 1 (9,3)[128X[104X
    [4X[25Xgap>[125X [27XDisplayInformationPerfectGroups( 30720, 6 );[127X[104X
    [4X[28X#I Perfect group 30720:  A5 ( 2^4 x 2^4 ) C N 2^1[128X[104X
    [4X[28X#I   centre = 2  size = 2^11*3*5  orbit size = 384[128X[104X
    [4X[28X#I   Holt-Plesken class 1 (9,6)[128X[104X
    [4X[25Xgap>[125X [27XDisplayInformationPerfectGroups( Factorial( 8 ) / 2 );[127X[104X
    [4X[28X#I Perfect group 20160.1:  A5 x L3(2) 2^1[128X[104X
    [4X[28X#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 5 + 16[128X[104X
    [4X[28X#I   Holt-Plesken class 31 (1,1) (occurs also in class 32)[128X[104X
    [4X[28X#I Perfect group 20160.2:  A5 2^1 x L3(2)[128X[104X
    [4X[28X#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 7 + 24[128X[104X
    [4X[28X#I   Holt-Plesken class 31 (1,2) (occurs also in class 32)[128X[104X
    [4X[28X#I Perfect group 20160.3:  ( A5 x L3(2) ) 2^1[128X[104X
    [4X[28X#I   centre = 2  size = 2^6*3^2*5*7  orbit size = 192[128X[104X
    [4X[28X#I   Holt-Plesken class 31 (1,3)[128X[104X
    [4X[28X#I Perfect group 20160.4:  simple group  A8[128X[104X
    [4X[28X#I   size = 2^6*3^2*5*7  orbit size = 8[128X[104X
    [4X[28X#I   Holt-Plesken class 26 (0,1)[128X[104X
    [4X[28X#I Perfect group 20160.5:  simple group  L3(4)[128X[104X
    [4X[28X#I   size = 2^6*3^2*5*7  orbit size = 21[128X[104X
    [4X[28X#I   Holt-Plesken class 27 (0,1)[128X[104X
  [4X[32X[104X
  
  
  [1X50.6-7 [33X[0;0YMore about the Perfect Groups Library[133X[101X
  
  [33X[0;0YFor  any  library  group  [22XG[122X,  the  library  files  do  not  only  provide  a
  presentation,  but,  in  addition, a list of one or more subgroups [22XS_1, ...,
  S_r[122X  of  [22XG[122X  such that there is a faithful permutation representation of [22XG[122X of
  degree  [22X∑_{i  =  1}^r [G:S_i][122X on the set [22X{ S_i g ∣ 1 ≤ i ≤ r, g ∈ G }[122X of the
  cosets  of  the  [22XS_i[122X. This allows one to construct the groups as permutation
  groups.  The function [2XDisplayInformationPerfectGroups[102X ([14X50.6-6[114X) displays only
  the available degree. The message[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xorbit size = 8[128X[104X
  [4X[32X[104X
  
  [33X[0;0Yin  the above example means that the available permutation representation is
  transitive and of degree 8, whereas the message[133X
  
  [4X[32X  Example  [32X[104X
    [4X[28Xorbit sizes = 5 + 16[128X[104X
  [4X[32X[104X
  
  [33X[0;0Ymeans  that  a  nontransitive  permutation representation is available which
  acts on two orbits of size 5 and 16 respectively.[133X
  
  [33X[0;0YThe   notation   used  in  the  [21Xdescription[121X  of  a  group  is  explained  in
  section 5.1.2 of [HP89]. We quote the respective page from there:[133X
  
  [33X[0;0YWithin  a  class  [22XQ#p[122X,  an  isomorphism type of groups will be denoted by an
  ordered  pair  of integers [22X(r,n)[122X, where [22Xr ≥ 0[122X and [22Xn > 0[122X. More precisely, the
  isomorphism types in [22XQ # p[122X of order [22Xp^r |Q|[122X will be denoted by [22X(r,1), (r,2),
  (r,3), ...[122X. Thus [22XQ[122X will always get the size number [22X(0,1)[122X.[133X
  
  [33X[0;0YIn addition to the symbol [22X(r,n)[122X, the groups in [22XQ#p[122X will also be given a more
  descriptive name. The purpose of this is to provide a very rough idea of the
  structure of the group. The names are derived in the following manner. First
  of all, the isomorphism classes of irreducible [22XF_pQ[122X-modules [22XM[122X with [22X|Q|.|M| ≤
  10^6[122X,  where  [22XF_p[122X  is the field of order [22Xp[122X, are assigned symbols. These will
  either  be  simply [22Xp^x[122X, where [22Xx[122X is the dimension of the module, or, if there
  is  more  than  one isomorphism class of irreducible modules having the same
  dimension,  they  will  be  denoted  by[22Xp^x[122X, [22Xp^{x'}[122X, etc. The one-dimensional
  module with trivial [22XQ[122X-action will therefore be denoted by [22Xp^1[122X. These symbols
  will  be  listed  under  the  description  of  [22XQ[122X.  The  group  name consists
  essentially of a list of the composition factors working from the top of the
  group  downwards;  hence  it  always starts with the name of [22XQ[122X itself. (This
  convention  is  the most convenient in our context, but it is different from
  that  adopted  in the ATLAS [CCN+85], for example, where composition factors
  are  listed  in the reverse order. For example, we denote a group isomorphic
  to [22XSL(2,5)[122X by [22XA_5 2^1[122X rather than [22X2.A_5[122X.)[133X
  
  [33X[0;0YSome  other  symbols are used in the name, in order to give some idea of the
  relationship between these composition factors, and splitting properties. We
  shall now list these additional symbols.[133X
  
  [8X[22X×[122X[108X
        [33X[0;6Ybetween  two  factors  denotes  a  direct  product  of [22XF_pQ[122X-modules or
        groups.[133X
  
  [8XC[108X
        [33X[0;6Y(for [21Xcommutator[121X) between two factors means that the second lies in the
        commutator  subgroup  of  the  first. Similarly, a segment of the form
        [22X(f_1  ×  f_2)  C  f_3[122X  would mean that the factors [22Xf_1[122X and [22Xf_2[122X commute
        modulo [22Xf_3[122X and [22Xf_3[122X lies in [22X[f_1,f_2][122X.[133X
  
  [8XA[108X
        [33X[0;6Y(for  [21Xabelian[121X) between two factors indicates that the second is in the
        [22Xp[122Xth  power  (but not the commutator subgroup) of the first. [21XA[121X may also
        follow the factors, if bracketed.[133X
  
  [8XE[108X
        [33X[0;6Y(for  [21Xelementary  abelian[121X) between two factors indicates that together
        they generate an elementary abelian group (modulo subsequent factors),
        but that the resulting [22XF_p Q[122X-module extension does not split.[133X
  
  [8XN[108X
        [33X[0;6Y(for  [21Xnonsplit[121X)  before  a  factor  indicates  that [22XQ[122X (or possibly its
        covering  group)  splits  down  as far at this factor but not over the
        factor  itself.  So [21X[22XQ f_1 N f_2[122X[121X means that the normal subgroup [22Xf_1 f_2[122X
        of  the  group  has  no  complement  but, modulo [22Xf_2[122X, [22Xf_1[122X, does have a
        complement.[133X
  
  [33X[0;0YBrackets have their obvious meaning. Summarizing, we have:[133X
  
  [8X[22X×[122X[108X
        [33X[0;6Y= direct product;[133X
  
  [8XC[108X
        [33X[0;6Y= commutator subgroup;[133X
  
  [8XA[108X
        [33X[0;6Y= abelian;[133X
  
  [8XE[108X
        [33X[0;6Y= elementary abelian; and[133X
  
  [8XN[108X
        [33X[0;6Y= nonsplit.[133X
  
  [33X[0;0YHere are some examples.[133X
  
  [8X(i)[108X
        [33X[0;6Y[22XA_5  (2^4  E 2^1 E 2^4) A[122X means that the pairs [22X2^4 E 2^1[122X and [22X2^1 E 2^4[122X
        are both elementary abelian of exponent 4.[133X
  
  [8X(ii)[108X
        [33X[0;6Y[22XA_5  (2^4  E  2^1  A)  C  2^1[122X  means that [22XO_2(G)[122X is of symplectic type
        [22X2^{1+5}[122X, with Frattini factor group of type [22X2^4 E 2^1[122X. The [21XA[121X after the
        [22X2^1[122X  indicates that [22XG[122X has a central cyclic subgroup [22X2^1 A 2^1[122X of order
        4.[133X
  
  [8X(iii)[108X
        [33X[0;6Y[22XL_3(2)  ((2^1 E) × ( N 2^3 E 2^{3'} A) C) 2^{3'}[122X means that the [22X2^{3'}[122X
        factor at the bottom lies in the commutator subgroup of the pair [22X2^3 E
        2^{3'}[122X in the middle, but the lower pair [22X2^{3'} A 2^{3'}[122X is abelian of
        exponent  4.  There is also a submodule [22X2^1 E 2^{3'}[122X, and the covering
        group  [22XL_3(2) 2^1[122X of [22XL_3(2)[122X does not split over the [22X2^3[122X factor. (Since
        [22XG[122X  is  perfect,  it  goes without saying that the extension [22XL_3(2) 2^1[122X
        cannot split itself.)[133X
  
  [33X[0;0YWe  must  stress that this notation does not always succeed in being precise
  or even unambiguous, and the reader is free to ignore it if it does not seem
  helpful.[133X
  
  [33X[0;0YIf  such a group description has been given in the book for [22XG[122X (and, in fact,
  this  is  the  case  for  most  of  the  library groups), it is displayed by
  [2XDisplayInformationPerfectGroups[102X  ([14X50.6-6[114X). Otherwise the function provides a
  less  explicit description of the (in these cases unique) Holt-Plesken class
  to  which  [22XG[122X  belongs, together with a serial number if this is necessary to
  make it unique.[133X
  
  
  [1X50.7 [33X[0;0YIrreducible Maximal Finite Integral Matrix Groups[133X[101X
  
  [33X[0;0YA  library  of irreducible maximal finite integral matrix groups is provided
  with  [5XGAP[105X.  It  contains  [22Xℚ[122X-class representatives for all of these groups of
  dimension  at most 31, and [22Xℤ[122X-class representatives for those of dimension at
  most 11 or of dimension 13, 17, 19, or 23.[133X
  
  [33X[0;0YThe groups provided in this library have been determined by Wilhelm Plesken,
  partially  as  joint work with Michael Pohst, or by members of his institute
  (Lehrstuhl  B  für Mathematik, RWTH Aachen). In particular, the data for the
  groups  of  dimensions  2  to  9 have been taken from the output of computer
  calculations  which they performed in 1979 (see [PP77], [PP80]). The [22Xℤ[122X-class
  representatives  of  the  groups  of  dimension  10 have been determined and
  computed  by Bernd Souvignier ([Sou94]), and those of dimensions 11, 13, and
  17  have  been  recomputed for this library from the circulant Gram matrices
  given  in  [Ple85],  using  the  stand-alone programs for the computation of
  short  vectors  and  Bravais  groups  which have been developed in Plesken's
  institute. The [22Xℤ[122X-class representatives of the groups of dimensions 19 and 23
  had  already  been  determined in [Ple85]. Gabriele Nebe has recomputed them
  for  us.  Her  main  contribution  to this library, however, is that she has
  determined  and  computed  the  [22Xℚ[122X-class  representatives  of  the  groups of
  non-prime dimensions between 12 and 24 and the groups of dimensions 25 to 31
  (see [PN95], [NP95b], [Neb95], [Neb96]).[133X
  
  [33X[0;0YThe  library  has  been  brought  into  [5XGAP[105X format by Volkmar Felsch. He has
  applied  several  [5XGAP[105X  routines  to  check  certain consistency of the data.
  However,  the  credit  and  responsibility  for  the  lists  remain with the
  authors.  We  are  grateful  to  Wilhelm  Plesken,  Gabriele Nebe, and Bernd
  Souvignier for supplying their results to [5XGAP[105X.[133X
  
  [33X[0;0YIn  the  preceding acknowledgement, we used some notations that will also be
  needed in the sequel. We first define these.[133X
  
  [33X[0;0YAny  integral matrix group [22XG[122X of dimension [22Xn[122X is a subgroup of [22XGL_n(ℤ)[122X as well
  as  of  [22XGL_n(ℚ)[122X  and  hence  lies in some conjugacy class of integral matrix
  groups  under  [22XGL_n(ℤ)[122X  and  also in some conjugacy class of rational matrix
  groups  under  [22XGL_n(ℚ)[122X.  As usual, we call these classes the [22Xℤ[122X-class and the
  [22Xℚ[122X-class  of  [22XG[122X,  respectively. Note that any conjugacy class of subgroups of
  [22XGL_n(ℚ)[122X  contains at least one [22Xℤ[122X-class of subgroups of [22XGL_n(ℤ)[122X and hence can
  be considered as the [22Xℚ[122X-class of some integral matrix group.[133X
  
  [33X[0;0YIn  the  context  of  this  library we are only concerned with [22Xℤ[122X-classes and
  [22Xℚ[122X-classes  of  subgroups of [22XGL_n(ℤ)[122X which are irreducible and maximal finite
  in  [22XGL_n(ℤ)[122X  (we  will  call  them  [13Xi.m.f.[113X subgroups  of  [22XGL_n(ℤ)[122X).  We  can
  distinguish two types of these groups:[133X
  
  [33X[0;0YFirst,  there  are  those i.m.f. subgroups of [22XGL_n(ℤ)[122X which are also maximal
  finite  subgroups  of  [22XGL_n(ℚ)[122X.  Let us denote the set of their [22Xℚ[122X-classes by
  [22XQ_1(n)[122X.  It  is clear from the above remark that [22XQ_1(n)[122X just consists of the
  [22Xℚ[122X-classes of i.m.f. subgroups of [22XGL_n(ℚ)[122X.[133X
  
  [33X[0;0YSecondly,  there  is  the  set  [22XQ_2(n)[122X  of  the  [22Xℚ[122X-classes  of the remaining
  i.m.f. subgroups  of  [22XGL_n(ℤ)[122X,  i.e.,  of those which are not maximal finite
  subgroups  of [22XGL_n(ℚ)[122X. For any such group [22XG[122X, there is at least one class [22XC ∈
  Q_1(n)[122X such that [22XG[122X is conjugate under [22Xℚ[122X to a proper subgroup of some group [22XH
  ∈  C[122X.  In fact, the class [22XC[122X is uniquely determined for any group [22XG[122X occurring
  in  the  library  (though  there  seems  to be no reason to assume that this
  property  should  hold  in  general).  Hence  we  may  call  [22XC[122X  the [13Xrational
  i.m.f. class[113X  of  [22XG[122X. Finally, we will denote the number of classes in [22XQ_1(n)[122X
  and [22XQ_2(n)[122X by [22Xq_1(n)[122X and [22Xq_2(n)[122X, respectively.[133X
  
  [33X[0;0YAs  an  example,  let  us  consider the case [22Xn = 4[122X. There are 6 [22Xℤ[122X-classes of
  i.m.f. subgroups  of  [22XGL_4(ℤ)[122X with representative subgroups [22XG_1, ..., G_6[122X of
  isomorphism types [22XG_1 ≅ W(F_4)[122X, [22XG_2 ≅ D_12 ≀ C_2[122X, [22XG_3 ≅ G_4 ≅ C_2 × S_5[122X, [22XG_5
  ≅ W(B_4)[122X, and [22XG_6 ≅ (D_12[122X[10XY[110X[22XD_12) : C_2[122X. The corresponding [22Xℚ[122X-classes, which we
  denote  [22XR_1, ..., R_6[122X, are pairwise different except that [22XR_3[122X coincides with
  [22XR_4[122X.  The  groups [22XG_1[122X, [22XG_2[122X, and [22XG_3[122X are i.m.f. subgroups of [22XGL_4(ℚ)[122X, but [22XG_5[122X
  and [22XG_6[122X are not because they are conjugate under [22XGL_4(ℚ)[122X to proper subgroups
  of  [22XG_1[122X and [22XG_2[122X, respectively. So we have [22XQ_1(4) = { R_1, R_2, R_3 }[122X, [22XQ_2(4)
  = { R_5, R_6 }[122X, [22Xq_1(4) = 3[122X, and [22Xq_2(4) = 2[122X.[133X
  
  [33X[0;0YThe [22Xq_1(n)[122X [22Xℚ[122X-classes of i.m.f. subgroups of [22XGL_n(ℚ)[122X have been determined for
  each   dimension   [22Xn  ≤  31[122X.  The  current  [5XGAP[105X  library  provides  integral
  representative  groups  for  all  these  classes. Moreover, all [22Xℤ[122X-classes of
  i.m.f. subgroups  of [22XGL_n(ℤ)[122X are known for [22Xn ≤ 11[122X and for [22Xn ∈ {13,17,19,23}[122X.
  For  these dimensions, the library offers integral representative groups for
  all  [22Xℚ[122X-classes  in  [22XQ_1(n)[122X  and  [22XQ_2(n)[122X  as  well  as  for  all [22Xℤ[122X-classes of
  i.m.f. subgroups of [22XGL_n(ℤ)[122X.[133X
  
  [33X[0;0YAny  group  [22XG[122X  of  dimension  [22Xn[122X  given  in the library is represented as the
  automorphism  group [22XG = Aut(F,L) = { g ∈ GL_n(ℤ) ∣ Lg = L, g F g^tr = F }[122X of
  a positive definite symmetric [22Xn × n[122X matrix [22XF ∈ ℤ^{n × n}[122X on an [22Xn[122X-dimensional
  lattice  [22XL  ≅  ℤ^{1 × n}[122X (for details see e.g. [PN95]). [5XGAP[105X provides for [22XG[122X a
  list of matrix generators and the [13XGram matrix[113X [22XF[122X.[133X
  
  [33X[0;0YThe  positive  definite  quadratic form defined by [22XF[122X defines a [13Xnorm[113X [22Xv F v^tr[122X
  for  each vector [22Xv ∈ L[122X, and there is only a finite set of vectors of minimal
  norm.  These  vectors  are  often simply called the [13Xshort vectors[113X. Their set
  splits  into orbits under [22XG[122X, and [22XG[122X being irreducible acts faithfully on each
  of  these  orbits by multiplication from the right. [5XGAP[105X provides for each of
  these orbits the orbit size and a representative vector.[133X
  
  [33X[0;0YLike  most of the other [5XGAP[105X libraries, the library of i.m.f. integral matrix
  groups  supplies  an  extraction  function,  [10XImfMatrixGroup[110X. However, as the
  library involves only 525 different groups, there is no need for a selection
  or  an  example  function.  Instead,  there are two functions, [2XImfInvariants[102X
  ([14X50.7-3[114X)  and  [2XDisplayImfInvariants[102X  ([14X50.7-2[114X),  which  provide  some [22Xℤ[122X-class
  invariants   that  can  be  extracted  from  the  library  without  actually
  constructing  the  representative  groups themselves. The difference between
  these  two  functions  is that the latter one displays the resulting data in
  some  easily  readable  format, whereas the first one returns them as record
  components so that you can properly access them.[133X
  
  [33X[0;0YWe  shall  give  an individual description of each of the library functions,
  but first we would like to insert a short remark concerning their names: Any
  self-explaining  name  of  a  function  handling  [13Xirreducible maximal finite
  integral  matrix  groups[113X  would have to include this term in full length and
  hence  would  grow  extremely  long.  Therefore  we  have decided to use the
  abbreviation  [10XImf[110X  instead in order to restrict the names to some reasonable
  length.[133X
  
  [33X[0;0YThe first three functions can be used to formulate loops over the classes.[133X
  
  [1X50.7-1 ImfNumberQQClasses[101X
  
  [33X[1;0Y[29X[2XImfNumberQQClasses[102X( [3Xdim[103X ) [32X function[133X
  [33X[1;0Y[29X[2XImfNumberQClasses[102X( [3Xdim[103X ) [32X function[133X
  [33X[1;0Y[29X[2XImfNumberZClasses[102X( [3Xdim[103X, [3Xq[103X ) [32X function[133X
  
  [33X[0;0Y[10XImfNumberQQClasses[110X   returns   the   number   [22Xq_1([122X[3Xdim[103X[22X)[122X   of   [22Xℚ[122X-classes   of
  i.m.f. rational  matrix groups of dimension [3Xdim[103X. Valid values of [3Xdim[103X are all
  positive integers up to 31.[133X
  
  [33X[0;0YNote:  In  order  to  enable  you to loop just over the classes belonging to
  [22XQ_1([122X[3Xdim[103X[22X)[122X,  we  have  arranged the list of [22Xℚ[122X-classes of dimension [3Xdim[103X for any
  dimension [3Xdim[103X in the library such that, whenever the classes of [22XQ_2([122X[3Xdim[103X[22X)[122X are
  known,  too, i.e., in the cases [22Xdim ≤ 11[122X or [22Xdim ∈ {13,17,19,23}[122X, the classes
  of  [22XQ_1([122X[3Xdim[103X[22X)[122X  precede  those  of  [22XQ_2([122X[3Xdim[103X[22X)[122X  and hence are numbered from 1 to
  [22Xq_1([122X[3Xdim[103X[22X)[122X.[133X
  
  [33X[0;0Y[10XImfNumberQClasses[110X returns the number of [22Xℚ[122X-classes of groups of dimension [3Xdim[103X
  which are available in the library. If [22Xdim ≤ 11[122X or [22Xdim ∈ {13,17,19,23}[122X, this
  is  the  number  [22Xq_1([122X[3Xdim[103X[22X)  +  q_2([122X[3Xdim[103X[22X)[122X  of  [22Xℚ[122X-classes of i.m.f. subgroups of
  [22XGL_dim(ℤ)[122X.  Otherwise,  it  is  just  the  number  [22Xq_1([122X[3Xdim[103X[22X)[122X  of [22Xℚ[122X-classes of
  i.m.f. subgroups of [22XGL_dim(ℚ)[122X. Valid values of [3Xdim[103X are all positive integers
  up to 31.[133X
  
  [33X[0;0Y[2XImfNumberZClasses[102X  returns  the  number  of [22Xℤ[122X-classes in the [3Xq[103X-th [22Xℚ[122X-class of
  i.m.f. integral  matrix groups of dimension [3Xdim[103X. Valid values of [3Xdim[103X are all
  positive integers up to 11 and all primes up to 23.[133X
  
  [1X50.7-2 DisplayImfInvariants[101X
  
  [33X[1;0Y[29X[2XDisplayImfInvariants[102X( [3Xdim[103X, [3Xq[103X[, [3Xz[103X] ) [32X function[133X
  
  [33X[0;0Y[2XDisplayImfInvariants[102X displays the following [22Xℤ[122X-class invariants of the groups
  in  the [3Xz[103X-th [22Xℤ[122X-class in the [3Xq[103X-th [22Xℚ[122X-class of i.m.f. integral matrix groups of
  dimension [3Xdim[103X:[133X
  
  [30X    [33X[0;6Yits  [22Xℤ[122X-class  number  in  the  form [3Xdim[103X.[3Xq[103X.[3Xz[103X, if [3Xdim[103X is at most 11 or a
        prime at most 23, or its [22Xℚ[122X-class number in the form [3Xdim[103X.[3Xq[103X, else,[133X
  
  [30X    [33X[0;6Ya message if the group is solvable,[133X
  
  [30X    [33X[0;6Ythe size of the group,[133X
  
  [30X    [33X[0;6Ythe isomorphism type of the group,[133X
  
  [30X    [33X[0;6Ythe elementary divisors of the associated quadratic form,[133X
  
  [30X    [33X[0;6Ythe  sizes of the orbits of short vectors (these sizes are the degrees
        of  the  faithful  permutation representations which you may construct
        using     the     functions     [2XIsomorphismPermGroup[102X    ([14X50.7-5[114X)    or
        [2XIsomorphismPermGroupImfGroup[102X ([14X50.7-6[114X) below),[133X
  
  [30X    [33X[0;6Ythe norm of the associated short vectors,[133X
  
  [30X    [33X[0;6Yonly  in  case  that  the  group is not an i.m.f. group in [22XGL_n(ℚ)[122X: an
        appropriate message, including the [22Xℚ[122X-class number of the corresponding
        rational i.m.f. class.[133X
  
  [33X[0;0YIf  you  specify  the  value  0  for any of the parameters [3Xdim[103X, [3Xq[103X, or [3Xz[103X, the
  command  will  loop  over  all  available  dimensions,  [22Xℚ[122X-classes  of  given
  dimension,  or  [22Xℤ[122X-classes within the given [22Xℚ[122X-class, respectively. Otherwise,
  the  values  of  the  arguments  must be in range. A value [3Xz[103X [22X≠ 1[122X must not be
  specified  if  the [22Xℤ[122X-classes are not known for the given dimension, i.e., if
  [3Xdim[103X [22X> 11[122X and [3Xdim[103X [22Xnot ∈ { 13, 17, 19, 23 }[122X. The default value of [3Xz[103X is 1. This
  value  of  [3Xz[103X  will  be accepted even if the [22Xℤ[122X-classes are not known. Then it
  specifies  the  only  representative  group  which is available for the [3Xq[103X-th
  [22Xℚ[122X-class. The greatest legal value of [3Xdim[103X is 31.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XDisplayImfInvariants( 3, 1, 0 );[127X[104X
    [4X[28X#I Z-class 3.1.1:  Solvable, size = 2^4*3[128X[104X
    [4X[28X#I   isomorphism type = C2 wr S3 = C2 x S4 = W(B3)[128X[104X
    [4X[28X#I   elementary divisors = 1^3[128X[104X
    [4X[28X#I   orbit size = 6, minimal norm = 1[128X[104X
    [4X[28X#I Z-class 3.1.2:  Solvable, size = 2^4*3[128X[104X
    [4X[28X#I   isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)[128X[104X
    [4X[28X#I   elementary divisors = 1*4^2[128X[104X
    [4X[28X#I   orbit size = 8, minimal norm = 3[128X[104X
    [4X[28X#I Z-class 3.1.3:  Solvable, size = 2^4*3[128X[104X
    [4X[28X#I   isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)[128X[104X
    [4X[28X#I   elementary divisors = 1^2*4[128X[104X
    [4X[28X#I   orbit size = 12, minimal norm = 2[128X[104X
    [4X[25Xgap>[125X [27XDisplayImfInvariants( 8, 15, 1 );[127X[104X
    [4X[28X#I Z-class 8.15.1:  Solvable, size = 2^5*3^4[128X[104X
    [4X[28X#I   isomorphism type = C2 x (S3 wr S3)[128X[104X
    [4X[28X#I   elementary divisors = 1*3^3*9^3*27[128X[104X
    [4X[28X#I   orbit size = 54, minimal norm = 8[128X[104X
    [4X[28X#I   not maximal finite in GL(8,Q), rational imf class is 8.5[128X[104X
    [4X[25Xgap>[125X [27XDisplayImfInvariants( 20, 23 );[127X[104X
    [4X[28X#I Q-class 20.23:  Size = 2^5*3^2*5*11[128X[104X
    [4X[28X#I   isomorphism type = (PSL(2,11) x D12).C2[128X[104X
    [4X[28X#I   elementary divisors = 1^18*11^2[128X[104X
    [4X[28X#I   orbit size = 3*660 + 2*1980 + 2640 + 3960, minimal norm = 4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YNote  that  the  function  [2XDisplayImfInvariants[102X  uses a kind of shorthand to
  display  the  elementary divisors. E. g., the expression [10X1*3^3*9^3*27[110X in the
  preceding  example stands for the elementary divisors [22X1,3,3,3,9,9,9,27[122X. (See
  also  the  next example which shows that the function [2XImfInvariants[102X ([14X50.7-3[114X)
  provides the elementary divisors in form of an ordinary [5XGAP[105X list.)[133X
  
  [33X[0;0YIn  the  description  of  the  isomorphism types the following notations are
  used:[133X
  
  [8X[22XA[122X [10Xx[110X[8X [22XB[122X [108X
        [33X[0;6Ydenotes a direct product of a group [22XA[122X by a group [22XB[122X,[133X
  
  [8X[22XA[122X [10Xsubd[110X[8X [22XB[122X [108X
        [33X[0;6Ydenotes a subdirect product of [22XA[122X by [22XB[122X,[133X
  
  [8X[22XA[122X [10XY[110X[8X [22XB[122X [108X
        [33X[0;6Ydenotes a central product of [22XA[122X by [22XB[122X,[133X
  
  [8X[22XA[122X [10Xwr[110X[8X [22XB[122X [108X
        [33X[0;6Ydenotes a wreath product of [22XA[122X by [22XB[122X,[133X
  
  [8X[22XA[122X[10X:[110X[8X[22XB[122X [108X
        [33X[0;6Ydenotes a split extension of [22XA[122X by [22XB[122X,[133X
  
  [8X[22XA[122X[10X.[110X[8X[22XB[122X [108X
        [33X[0;6Ydenotes just an extension of [22XA[122X by [22XB[122X (split or nonsplit).[133X
  
  [33X[0;0YThe groups involved are[133X
  
  [30X    [33X[0;6Ythe cyclic groups [22XC_n[122X, dihedral groups [22XD_n[122X, and generalized quaternion
        groups [22XQ_n[122X of order [22Xn[122X, denoted by [10XC[110X[3Xn[103X, [10XD[110X[3Xn[103X, and [10XQ[110X[3Xn[103X, respectively,[133X
  
  [30X    [33X[0;6Ythe  alternating  groups  [22XA_n[122X  and  symmetric  groups [22XS_n[122X of degree [22Xn[122X,
        denoted by [10XA[110X[3Xn[103X and [10XS[110X[3Xn[103X, respectively,[133X
  
  [30X    [33X[0;6Ythe linear groups [22XGL_n(q)[122X, [22XPGL_n(q)[122X, [22XSL_n(q)[122X, and [22XPSL_n(q)[122X, denoted by
        [10XGL[110X([3Xn[103X,[3Xq[103X), [10XPGL[110X([3Xn[103X,[3Xq[103X), [10XSL[110X([3Xn[103X,[3Xq[103X), and [10XPSL[110X([3Xn[103X,[3Xq[103X), respectively,[133X
  
  [30X    [33X[0;6Ythe  unitary  groups  [22XSU_n(q)[122X  and  [22XPSU_n(q)[122X,  denoted  by [10XSU[110X([3Xn[103X,[3Xq[103X) and
        [10XPSU[110X([3Xn[103X,[3Xq[103X), respectively,[133X
  
  [30X    [33X[0;6Ythe  symplectic  groups  [22XSp(n,q)[122X  and [22XPSp(n,q)[122X, denoted by [10XSp[110X([3Xn[103X,[3Xq[103X) and
        [10XPSp[110X([3Xn[103X,[3Xq[103X), respectively,[133X
  
  [30X    [33X[0;6Ythe  orthogonal  groups [22XO_8^+(2)[122X and [22XPO_8^+(2)[122X, denoted by [10XO+[110X(8,2) and
        [10XPO+[110X(8,2), respectively,[133X
  
  [30X    [33X[0;6Ythe   extraspecial   groups   [22X2_+^{1+8}[122X,   [22X3_+^{1+2}[122X,  [22X3_+^{1+4}[122X,  and
        [22X5_+^{1+2}[122X,  denoted  by  [10X2+^(1+8)[110X,  [10X3+^(1+2)[110X,  [10X3+^(1+4)[110X, and [10X5+^(1+2)[110X,
        respectively,[133X
  
  [30X    [33X[0;6Ythe Chevalley group [22XG_2(3)[122X, denoted by [10XG2(3)[110X,[133X
  
  [30X    [33X[0;6Ythe twisted Chevalley group [22X^3D_4(2)[122X, denoted by [10X3D4(2)[110X,[133X
  
  [30X    [33X[0;6Ythe Suzuki group [22XSz(8)[122X, denoted by [10XSz(8)[110X,[133X
  
  [30X    [33X[0;6Ythe Weyl groups [22XW(A_n)[122X, [22XW(B_n)[122X, [22XW(D_n)[122X, [22XW(E_n)[122X, and [22XW(F_4)[122X, denoted by
        [10XW(A[3Xn[103X[10X)[110X, [10XW(B[3Xn[103X[10X)[110X, [10XW(D[3Xn[103X[10X)[110X, [10XW(E[3Xn[103X[10X)[110X, and [10XW(F4)[110X, respectively,[133X
  
  [30X    [33X[0;6Ythe  sporadic  simple  groups  [22XCo_1[122X,  [22XCo_2[122X, [22XCo_3[122X, [22XHS[122X, [22XJ_2[122X, [22XM_12[122X, [22XM_22[122X,
        [22XM_23[122X,  [22XM_24[122X,  and [22XMc[122X, denoted by [10XCo1[110X, [10XCo2[110X, [10XCo3[110X, [10XHS[110X, [10XJ2[110X, [10XM12[110X, [10XM22[110X, [10XM23[110X,
        [10XM24[110X, and [10XMc[110X, respectively,[133X
  
  [30X    [33X[0;6Ya point stabilizer of index 11 in [22XM_11[122X, denoted by [10XM10[110X.[133X
  
  [33X[0;0YAs  mentioned above, the data assembled by the function [2XDisplayImfInvariants[102X
  are [21Xcheap data[121X in the sense that they can be provided by the library without
  loading any of its large matrix files or performing any matrix calculations.
  The  following  function allows you to get proper access to these cheap data
  instead of just displaying them.[133X
  
  [1X50.7-3 ImfInvariants[101X
  
  [33X[1;0Y[29X[2XImfInvariants[102X( [3Xdim[103X, [3Xq[103X[, [3Xz[103X] ) [32X function[133X
  
  [33X[0;0Y[2XImfInvariants[102X returns a record which provides some [22Xℤ[122X-class invariants of the
  groups  in  the  [3Xz[103X-th  [22Xℤ[122X-class in the [3Xq[103X-th [22Xℚ[122X-class of i.m.f. integral matrix
  groups  of  dimension  [3Xdim[103X.  A  value  [3Xz[103X  [22X≠  1[122X  must not be specified if the
  [22Xℤ[122X-classes  are  not known for the given dimension, i.e., if [3Xdim[103X [22X> 11[122X and [3Xdim[103X
  [22Xnot  ∈ { 13, 17, 19, 23 }[122X. The default value of [3Xz[103X is 1. This value of [3Xz[103X will
  be  accepted even if the [22Xℤ[122X-classes are not known. Then it specifies the only
  representative  group  which is available for the [3Xq[103X-th [22Xℚ[122X-class. The greatest
  legal value of [3Xdim[103X is 31.[133X
  
  [33X[0;0YThe resulting record contains six or seven components:[133X
  
  [8X[10Xsize[110X[8X [108X
        [33X[0;6Ythe size of any representative group [3XG[103X,[133X
  
  [8X[10XisSolvable[110X[8X [108X
        [33X[0;6Yis [9Xtrue[109X if [3XG[103X is solvable,[133X
  
  [8X[10XisomorphismType[110X[8X [108X
        [33X[0;6Ya  text  string  describing  the  isomorphism  type  of [3XG[103X (in the same
        notation as used by the function [10XDisplayImfInvariants[110X above),[133X
  
  [8X[10XelementaryDivisors[110X[8X [108X
        [33X[0;6Ythe  elementary  divisors of the associated Gram matrix [3XF[103X (in the same
        format as the result of the function [2XElementaryDivisorsMat[102X ([14X24.9-1[114X),[133X
  
  [8X[10XminimalNorm[110X[8X [108X
        [33X[0;6Ythe norm of the associated short vectors,[133X
  
  [8X[10XsizesOrbitsShortVectors[110X[8X [108X
        [33X[0;6Ythe sizes of the orbits of short vectors under [3XF[103X,[133X
  
  [8X[10XmaximalQClass[110X[8X [108X
        [33X[0;6Ythe  [22Xℚ[122X-class number of an i.m.f. group in [22XGL_n(ℚ)[122X that contains [3XG[103X as a
        subgroup  (only  in  case  that  not [3XG[103X itself is an i.m.f. subgroup of
        [22XGL_n(ℚ)[122X).[133X
  
  [33X[0;0YNote  that  four  of these data, namely the group size, the solvability, the
  isomorphism  type, and the corresponding rational i.m.f. class, are not only
  [22Xℤ[122X-class invariants, but also [22Xℚ[122X-class invariants.[133X
  
  [33X[0;0YNote  further  that, though the isomorphism type is a [22Xℚ[122X-class invariant, you
  will  sometimes  get  different  descriptions for different [22Xℤ[122X-classes of the
  same  [22Xℚ[122X-class (as, e.g., for the classes 3.1.1 and 3.1.2 in the last example
  above).  The  purpose  of this behaviour is to provide some more information
  about the underlying lattices.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XImfInvariants( 8, 15, 1 );[127X[104X
    [4X[28Xrec( elementaryDivisors := [ 1, 3, 3, 3, 9, 9, 9, 27 ], [128X[104X
    [4X[28X  isSolvable := true, isomorphismType := "C2 x (S3 wr S3)", [128X[104X
    [4X[28X  maximalQClass := 5, minimalNorm := 8, size := 2592, [128X[104X
    [4X[28X  sizesOrbitsShortVectors := [ 54 ] )[128X[104X
    [4X[25Xgap>[125X [27XImfInvariants( 24, 1 ).size;[127X[104X
    [4X[28X10409396852733332453861621760000[128X[104X
    [4X[25Xgap>[125X [27XImfInvariants( 23, 5, 2 ).sizesOrbitsShortVectors;[127X[104X
    [4X[28X[ 552, 53130 ][128X[104X
    [4X[25Xgap>[125X [27Xfor i in [ 1 .. ImfNumberQClasses( 22 ) ] do[127X[104X
    [4X[25X>[125X [27X   Print( ImfInvariants( 22, i ).isomorphismType, "\n" ); od;[127X[104X
    [4X[28XC2 wr S22 = W(B22)[128X[104X
    [4X[28X(C2 x PSU(6,2)).S3[128X[104X
    [4X[28X(C2 x S3) wr S11 = (C2 x W(A2)) wr S11[128X[104X
    [4X[28X(C2 x S12) wr C2 = (C2 x W(A11)) wr C2[128X[104X
    [4X[28XC2 x S3 x S12 = C2 x W(A2) x W(A11)[128X[104X
    [4X[28X(C2 x HS).C2[128X[104X
    [4X[28X(C2 x Mc).C2[128X[104X
    [4X[28XC2 x S23 = C2 x W(A22)[128X[104X
    [4X[28XC2 x PSL(2,23)[128X[104X
    [4X[28XC2 x PSL(2,23)[128X[104X
    [4X[28XC2 x PGL(2,23)[128X[104X
    [4X[28XC2 x PGL(2,23)[128X[104X
  [4X[32X[104X
  
  [1X50.7-4 ImfMatrixGroup[101X
  
  [33X[1;0Y[29X[2XImfMatrixGroup[102X( [3Xdim[103X, [3Xq[103X[, [3Xz[103X] ) [32X function[133X
  
  [33X[0;0Y[2XImfMatrixGroup[102X  is  the  essential extraction function of this library (note
  that  its  name has been changed from [10XImfMatGroup[110X in [5XGAP[105X 3 to [2XImfMatrixGroup[102X
  in  [5XGAP[105X 4). It returns a representative group, [22XG[122X say, of the [3Xz[103X-th [22Xℤ[122X-class in
  the  [3Xq[103X-th [22Xℚ[122X-class of i.m.f. integral matrix groups of dimension [3Xdim[103X. A value
  [3Xz[103X[22X≠  1[122X  must  not  be  specified if the [22Xℤ[122X-classes are not known for the given
  dimension,  i.e.,  if [3Xdim[103X [22X> 11[122X and [3Xdim[103X [22Xnot ∈ { 13, 17, 19, 23 }[122X. The default
  value  of [3Xz[103X is 1. This value of [3Xz[103X will be accepted even if the [22Xℤ[122X-classes are
  not  known.  Then  it  specifies  the  only  representative  group  which is
  available for the [3Xq[103X-th [22Xℚ[122X-class. The greatest legal value of [3Xdim[103X is 31.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XG := ImfMatrixGroup( 5, 1, 3 );[127X[104X
    [4X[28XImfMatrixGroup(5,1,3)[128X[104X
    [4X[25Xgap>[125X [27Xfor m in GeneratorsOfGroup( G ) do PrintArray( m ); od;[127X[104X
    [4X[28X[ [  -1,   0,   0,   0,   0 ],[128X[104X
    [4X[28X  [   0,   1,   0,   0,   0 ],[128X[104X
    [4X[28X  [   0,   0,   0,   1,   0 ],[128X[104X
    [4X[28X  [  -1,  -1,  -1,  -1,   2 ],[128X[104X
    [4X[28X  [  -1,   0,   0,   0,   1 ] ][128X[104X
    [4X[28X[ [  0,  1,  0,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  1,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  0,  1,  0 ],[128X[104X
    [4X[28X  [  1,  0,  0,  0,  0 ],[128X[104X
    [4X[28X  [  0,  0,  0,  0,  1 ] ][128X[104X
  [4X[32X[104X
  
  [33X[0;0YThe  attributes  [2XSize[102X  ([14X30.4-6[114X)  and  [10XIsSolvable[110X will be properly set in the
  resulting   matrix   group   [22XG[122X.   In   addition,   it   has  two  attributes
  [10XIsImfMatrixGroup[110X  and  [10XImfRecord[110X  where the first one is just a logical flag
  set  to  [9Xtrue[109X  and the latter one is a record. Except for the group size and
  the  solvability  flag,  this  record  contains  the  same components as the
  resulting  record  of  the  function [2XImfInvariants[102X ([14X50.7-3[114X) described above,
  namely  the components [10XisomorphismType[110X, [10XelementaryDivisors[110X, [10XminimalNorm[110X, and
  [10XsizesOrbitsShortVectors[110X   and,   if   [22XG[122X  is  not  a  rational  i.m.f. group,
  [10XmaximalQClass[110X. Moreover, it has the two components[133X
  
  [8X[10Xform[110X[8X[108X
        [33X[0;6Ythe associated Gram matrix [22XF[122X, and[133X
  
  [8X[10XrepsOrbitsShortVectors[110X[8X[108X
        [33X[0;6Yrepresentatives of the orbits of short vectors under [22XF[122X.[133X
  
  [33X[0;0YThe  last  one  of  these  components  will  be  required  by  the  function
  [2XIsomorphismPermGroup[102X ([14X50.7-5[114X) below.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XSize( G );[127X[104X
    [4X[28X3840[128X[104X
    [4X[25Xgap>[125X [27Ximf := ImfRecord( G );;[127X[104X
    [4X[25Xgap>[125X [27Ximf.isomorphismType;[127X[104X
    [4X[28X"C2 wr S5 = C2 x W(D5)"[128X[104X
    [4X[25Xgap>[125X [27XPrintArray( imf.form );[127X[104X
    [4X[28X[ [  4,  0,  0,  0,  2 ],[128X[104X
    [4X[28X  [  0,  4,  0,  0,  2 ],[128X[104X
    [4X[28X  [  0,  0,  4,  0,  2 ],[128X[104X
    [4X[28X  [  0,  0,  0,  4,  2 ],[128X[104X
    [4X[28X  [  2,  2,  2,  2,  5 ] ][128X[104X
    [4X[25Xgap>[125X [27Ximf.elementaryDivisors;[127X[104X
    [4X[28X[ 1, 4, 4, 4, 4 ][128X[104X
    [4X[25Xgap>[125X [27Ximf.minimalNorm;[127X[104X
    [4X[28X4[128X[104X
  [4X[32X[104X
  
  [33X[0;0YIf  you  want to perform calculations in such a matrix group [22XG[122X you should be
  aware  of  the fact that the permutation group routines of [5XGAP[105X are much more
  efficient  than  the  matrix  group routines. Hence we recommend that you do
  your  computations,  whenever  possible, in the isomorphic permutation group
  which  is  induced by the action of [22XG[122X on one of the orbits of the associated
  short   vectors.   You   may   call   one   of   the   following   functions
  [2XIsomorphismPermGroup[102X  ([14X50.7-5[114X)  or  [2XIsomorphismPermGroupImfGroup[102X ([14X50.7-6[114X) to
  get  an  isomorphism  to  such  a  permutation  group (note that these [5XGAP[105X 4
  functions    have    replaced    the    [5XGAP[105X 3    functions   [10XPermGroup[110X   and
  [10XPermGroupImfGroup[110X).[133X
  
  [1X50.7-5 IsomorphismPermGroup[101X
  
  [33X[1;0Y[29X[2XIsomorphismPermGroup[102X( [3XG[103X ) [32X method[133X
  
  [33X[0;0Yreturns an isomorphism, [22Xφ[122X say, from the given i.m.f. integral matrix group [22XG[122X
  to  a permutation group [22XP := φ(G)[122X acting on a minimal orbit, [22XS[122X say, of short
  vectors  of  [22XG[122X  such  that  each  matrix  [22Xm ∈ G[122X is mapped to the permutation
  induced by its action on [22XS[122X.[133X
  
  [33X[0;0YNote  that  in  case of a large orbit the construction of [22Xφ[122X may be space and
  time consuming. Fortunately, there are only six [22Xℚ[122X-classes in the library for
  which the smallest orbit of short vectors is of size greater than [22X20000[122X, the
  worst case being the orbit of size [22X196560[122X for the Leech lattice ([3Xdim[103X [22X= 24[122X, [3Xq[103X
  [22X= 3[122X).[133X
  
  [33X[0;0YThe  inverse  isomorphism [22Xφ^{-1}[122X from [22XP[122X to [22XG[122X is constructed by determining a
  [22Xℚ[122X-base  [22XB  ⊂  S[122X  of  [22Xℚ^{1  × dim}[122X in [22XS[122X and, in addition, the associated base
  change  matrix  [22XM[122X  which transforms [22XB[122X into the standard base of [22Xℤ^{1 × dim}[122X.
  This   allows  a  simple  computation  of  the  preimage  [22Xφ^{-1}(p)[122X  of  any
  permutation  [22Xp  ∈  P[122X,  as  follows. If, for [22X1 ≤ i ≤[122X [3Xdim[103X, [22Xb_i[122X is the position
  number  in [22XS[122X of the [22Xi[122X-th base vector in [22XB[122X, it suffices to look up the vector
  whose  position number in [22XS[122X is the image of [22Xb_i[122X under [22Xp[122X and to multiply this
  vector by [22XM[122X to get the [22Xi[122X-th row of [22Xφ^{-1}(p)[122X.[133X
  
  [33X[0;0YYou  may  use  the  functions [2XImage[102X ([14X32.4-6[114X) and [2XPreImage[102X ([14X32.5-6[114X) to switch
  from [22XG[122X to [22XP[122X and back from [22XP[122X to [22XG[122X.[133X
  
  [33X[0;0YAs  an  example,  let  us  continue  the  preceding  example and compute the
  solvable residuum of the group [22XG[122X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27X# Perform the computations in an isomorphic permutation group.[127X[104X
    [4X[25Xgap>[125X [27Xphi := IsomorphismPermGroup( G );;[127X[104X
    [4X[25Xgap>[125X [27XP := Image( phi );[127X[104X
    [4X[28XGroup([ (1,7,6)(2,9)(4,5,10), (2,3,4,5)(6,9,8,7) ])[128X[104X
    [4X[25Xgap>[125X [27XD := DerivedSubgroup( P );;[127X[104X
    [4X[25Xgap>[125X [27XSize( D );[127X[104X
    [4X[28X960[128X[104X
    [4X[25Xgap>[125X [27XIsPerfectGroup( D );[127X[104X
    [4X[28Xtrue[128X[104X
    [4X[25Xgap>[125X [27X# We have found the solvable residuum of P,[127X[104X
    [4X[25Xgap>[125X [27X# now move the results back to the matrix group G.[127X[104X
    [4X[25Xgap>[125X [27XR := PreImage( phi, D );;[127X[104X
    [4X[25Xgap>[125X [27XStructureDescription(R);[127X[104X
    [4X[28X"(C2 x C2 x C2 x C2) : A5"[128X[104X
    [4X[25Xgap>[125X [27XIdGroup(D)=IdGroup(R);[127X[104X
    [4X[28Xtrue[128X[104X
  [4X[32X[104X
  
  [1X50.7-6 IsomorphismPermGroupImfGroup[101X
  
  [33X[1;0Y[29X[2XIsomorphismPermGroupImfGroup[102X( [3XG[103X, [3Xn[103X ) [32X function[133X
  
  [33X[0;0Y[2XIsomorphismPermGroupImfGroup[102X  returns  an isomorphism, [22Xφ[122X say, from the given
  i.m.f. integral  matrix  group [3XG[103X to a permutation group [22XP[122X acting on the [3Xn[103X-th
  orbit, [22XS[122X say, of short vectors of [3XG[103X such that each matrix [22Xm ∈[122X [3XG[103X is mapped to
  the permutation induced by its action on [22XS[122X.[133X
  
  [33X[0;0YThe  only  difference to the above function [2XIsomorphismPermGroup[102X ([14X50.7-5[114X) is
  that  you  can specify the orbit to be used. In fact, as the orbits of short
  vectors are sorted by increasing sizes, the function [10XIsomorphismPermGroup( [3XG[103X[10X
  )[110X    has    been    implemented    such    that    it   is   equivalent   to
  [10XIsomorphismPermGroupImfGroup( [3XG[103X[10X, 1 )[110X.[133X
  
  [4X[32X  Example  [32X[104X
    [4X[25Xgap>[125X [27XImfInvariants( 12, 9 ).sizesOrbitsShortVectors;[127X[104X
    [4X[28X[ 120, 300 ][128X[104X
    [4X[25Xgap>[125X [27XG := ImfMatrixGroup( 12, 9 );[127X[104X
    [4X[28XImfMatrixGroup(12,9)[128X[104X
    [4X[25Xgap>[125X [27Xphi1 := IsomorphismPermGroupImfGroup( G, 1 );;[127X[104X
    [4X[25Xgap>[125X [27XP1 := Image( phi1 );[127X[104X
    [4X[28X<permutation group of size 2400 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XLargestMovedPoint( P1 );[127X[104X
    [4X[28X120[128X[104X
    [4X[25Xgap>[125X [27Xphi2 := IsomorphismPermGroupImfGroup( G, 2 );;[127X[104X
    [4X[25Xgap>[125X [27XP2 := Image( phi2 );[127X[104X
    [4X[28X<permutation group of size 2400 with 2 generators>[128X[104X
    [4X[25Xgap>[125X [27XLargestMovedPoint( P2 );[127X[104X
    [4X[28X300[128X[104X
  [4X[32X[104X
  
