48 Group Libraries

constructs a trivial group in the category given by the filter filter. If filter is not given it defaults to IsPcGroup.

gap> TrivialGroup();
<pc group of size 1 with 0 generators>
gap> TrivialGroup( IsPermGroup );
Group(())

constructs the cyclic group of size n in the category given by the filter filt. If filt is not given it defaults to IsPcGroup.

gap> CyclicGroup(12);
<pc group of size 12 with 3 generators>
gap> CyclicGroup(IsPermGroup,12);
Group([ (1,2,3,4,5,6,7,8,9,10,11,12) ])
gap> matgrp1:= CyclicGroup( IsMatrixGroup, 12 );
<matrix group of size 12 with 1 generators>
gap> FieldOfMatrixGroup( matgrp1 );
Rationals
gap> matgrp2:= CyclicGroup( IsMatrixGroup, GF(2), 12 );
<matrix group of size 12 with 1 generators>
gap> FieldOfMatrixGroup( matgrp2 );
GF(2)

constructs an abelian group in the category given by the filter filt which is of isomorphism type C_ints[1] * C_ints[2] * ¼* C_ints[n]. ints must be a list of positive integers. If filt is not given it defaults to IsPcGroup. The generators of the group returned are the elements corresponding to the integers in ints.

gap> AbelianGroup([1,2,3]);
<pc group of size 6 with 3 generators>

constructs the elementary abelian group of size n in the category given by the filter filt. If filt is not given it defaults to IsPcGroup.

gap> ElementaryAbelianGroup(8192);
<pc group of size 8192 with 13 generators>

constructs the dihedral group of size n in the category given by the filter filt. If filt is not given it defaults to IsPcGroup.

gap> DihedralGroup(10);
<pc group of size 10 with 2 generators>

Let order be of the form p²ⁿ⁺¹, for a prime integer p and a positive integer n. ExtraspecialGroup returns the extraspecial group of order order that is determined by exp, in the category given by the filter filt.

If p is odd then admissible values of exp are the exponent of the group (either p or p²) or one of '+', "+", '-', "-". For p = 2, only the above plus or minus signs are admissible.

If filt is not given it defaults to IsPcGroup.

gap> ExtraspecialGroup( 27, 3 );
<pc group of size 27 with 3 generators>
gap> ExtraspecialGroup( 27, '+' );
<pc group of size 27 with 3 generators>
gap> ExtraspecialGroup( 8, "-" );
<pc group of size 8 with 3 generators>

constructs the alternating group of degree deg in the category given by the filter filt. If filt is not given it defaults to IsPermGroup. In the second version, the function constructs the alternating group on the points given in the set dom which must be a set of positive integers.

gap> AlternatingGroup(5);
Alt( [ 1 .. 5 ] )

constructs the symmetric group of degree deg in the category given by the filter filt. If filt is not given it defaults to IsPermGroup. In the second version, the function constructs the symmetric group on the points given in the set dom which must be a set of positive integers.

gap> SymmetricGroup(10);
Sym( [ 1 .. 10 ] )

Note that permutation groups provide special treatment of symmetric and alternating groups, see Symmetric and Alternating Groups.

constructs the Mathieu group of degree degree in the category given by the filter filt, where degree must be in { 9, 10, 11, 12, 21, 22, 23, 24 }. If filt is not given it defaults to IsPermGroup.

gap> MathieuGroup( 11 );
Group([ (1,2,3,4,5,6,7,8,9,10,11), (3,7,11,8)(4,10,5,6) ])

Constructs a group isomorphic to the Suzuki group Sz( q ) over the field with q elements, where q ³ 8 is a non - square power of 2.

If filt is not given it defaults to IsMatrixGroup, and the returned group is the Suzuki group itself.

gap> SuzukiGroup( 32 );
Sz(32)

Constructs a group isomorphic to the Ree group 2G2(q) where q=3^1+2m for m > 0.

If filt is not given it defaults to IsMatrixGroup and the generating matrices are based on KLM01. (No particular choice of a generating set is guaranteed.)

gap> ReeGroup( 27 );
Ree(27)

48.2 Classical Groups

The following functions return classical groups. For the linear, symplectic, and unitary groups (the latter in dimension at least 3), the generators are taken from Tay87; for the unitary groups in dimension 2, the isomorphism of SU(2,q) and SL(2,q) is used, see for example Hup67. The generators of the orthogonal groups are taken from IshibashiEarnest94 and KleidmanLiebeck90, except that the generators of the orthogonal groups in odd dimension in even characteristic are constructed via the isomorphism to a symplectic group, see for example Car72a.

For symplectic and orthogonal matrix groups returned by the functions described below, the invariant bilinear form is stored as the value of the attribute InvariantBilinearForm (see InvariantBilinearForm). Analogously, the invariant sesquilinear form defining the unitary groups is stored as the value of the attribute InvariantSesquilinearForm (see InvariantSesquilinearForm). The defining quadratic form of orthogonal groups is stored as the value of the attribute InvariantQuadraticForm (see InvariantQuadraticForm).

The first two forms construct a group isomorphic to the general linear group GL( d, q ) of all d ×d matrices over the field with q elements, in the category given by the filter filt.

If filt is not given it defaults to IsMatrixGroup, and the returned group is the general linear group as a matrix group in its natural action (see also IsNaturalGL).

The third and the fourth form construct the group of all integer d ×d matrices invertible within the integers (see also IsNaturalGLnZ).

The last two forms construct the general linear group of degree d over the residue class ring Z/mZ.

gap> GL(4,3);
GL(4,3)
gap> GL(2,Integers);
GL(2,Integers)
gap> GL(3,Integers mod 12);
GL(3,Z/12Z)

The first two forms construct a group isomorphic to the special linear group SL( d, q ) of all those d ×d matrices over the field with q elements whose determinant is the identity of the field, in the category given by the filter filt.

If filt is not given it defaults to IsMatrixGroup, and the returned group is the special linear group as a matrix group in its natural action (see also IsNaturalSL).

The third and the fourth form construct the group of all integer d ×d matrices of determinant 1 (see also IsNaturalSLnZ).

The last two forms construct the group of all d ×d matrices over Z/mZ with determinant 1 (mod m).

gap> SpecialLinearGroup(2,2);
SL(2,2)
gap> SL(3,Integers);
SL(3,Integers)
gap> SL(4,Integers mod 4);
SL(4,Z/4Z)

Using the OnLines operation it is possible to obtain the corresponding projective groups in a permutation action:

gap> g:=GL(4,3);;Size(g);
24261120
gap> pgl:=Action(g,Orbit(g,Z(3)^0*[1,0,0,0],OnLines),OnLines);;
gap> Size(pgl);
12130560

constructs a group isomorphic to the general unitary group GU( d, q ) of those d ×d matrices over the field with q ² elements that respect a fixed nondegenerate sesquilinear form, in the category given by the filter filt.

If filt is not given it defaults to IsMatrixGroup, and the returned group is the general unitary group itself.

gap> GeneralUnitaryGroup( 3, 5 );
GU(3,5)

constructs a group isomorphic to the special unitary group GU(d, q) of those d ×d matrices over the field with q ² elements whose determinant is the identity of the field and that respect a fixed nondegenerate sesquilinear form, in the category given by the filter filt.

If filt is not given it defaults to IsMatrixGroup, and the returned group is the special unitary group itself.

gap> SpecialUnitaryGroup( 3, 5 );
SU(3,5)

constructs a group isomorphic to the symplectic group Sp( d, q ) of those d ×d matrices over the field with q elements that respect a fixed nondegenerate symplectic form, in the category given by the filter filt.

If filt is not given it defaults to IsMatrixGroup, and the returned group is the symplectic group itself.

gap> SymplecticGroup( 4, 2 );
Sp(4,2)

constructs a group isomorphic to the general orthogonal group GO( e, d, q ) of those d ×d matrices over the field with q elements that respect a non-singular quadratic form (see InvariantQuadraticForm) specified by e, in the category given by the filter filt.

The value of e must be 0 for odd d (and can optionally be omitted in this case), respectively one of 1 or -1 for even d. If filt is not given it defaults to IsMatrixGroup, and the returned group is the general orthogonal group itself.

Note that in KleidmanLiebeck90, GO is defined as the stabilizer D(V,F,k) of the quadratic form, up to scalars, whereas our GO is called I(V,F,k) there.

SpecialOrthogonalGroup returns a group isomorphic to the special orthogonal group SO( e, d, q ), which is the subgroup of all those matrices in the general orthogonal group (see GeneralOrthogonalGroup) that have determinant one, in the category given by the filter filt. (The index of SO( e, d, q ) in GO( e, d, q ) is 2 if q is odd, and 1 if q is even.)

If filt is not given it defaults to IsMatrixGroup, and the returned group is the special orthogonal group itself.

gap> GeneralOrthogonalGroup( 3, 7 );
GO(0,3,7)
gap> GeneralOrthogonalGroup( -1, 4, 3 );
GO(-1,4,3)
gap> SpecialOrthogonalGroup( 1, 4, 4 );
GO(+1,4,4)

constructs a group isomorphic to the projective general linear group PGL( d, q ) of those d ×d matrices over the field with q elements, modulo the centre, in the category given by the filter filt.

If filt is not given it defaults to IsPermGroup, and the returned group is the action on lines of the underlying vector space.

constructs a group isomorphic to the projective special linear group PSL( d, q ) of those d ×d matrices over the field with q elements whose determinant is the identity of the field, modulo the centre, in the category given by the filter filt.

If filt is not given it defaults to IsPermGroup, and the returned group is the action on lines of the underlying vector space.

constructs a group isomorphic to the projective general unitary group PGU( d, q ) of those d ×d matrices over the field with q ² elements that respect a fixed nondegenerate sesquilinear form, modulo the centre, in the category given by the filter filt.

If filt is not given it defaults to IsPermGroup, and the returned group is the action on lines of the underlying vector space.

constructs a group isomorphic to the projective special unitary group PSU( d, q ) of those d ×d matrices over the field with q ² elements that respect a fixed nondegenerate sesquilinear form and have determinant 1, modulo the centre, in the category given by the filter filt.

If filt is not given it defaults to IsPermGroup, and the returned group is the action on lines of the underlying vector space.

constructs a group isomorphic to the projective symplectic group PSp(d,q) of those d ×d matrices over the field with q elements that respect a fixed nondegenerate symplectic form, modulo the centre, in the category given by the filter filt.

If filt is not given it defaults to IsPermGroup, and the returned group is the action on lines of the underlying vector space.

48.3 Conjugacy Classes in Classical Groups

For general and special linear groups (see GeneralLinearGroup and SpecialLinearGroup) GAP 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.

gap> g := SL(4,9);
SL(4,9)
gap> NrConjugacyClasses(g);
861
gap> cl := ConjugacyClasses(g);;
gap> Length(cl);
861

The first of these functions compute for given n Î N and prime power q the number of conjugacy classes in the classical groups GL( n , q ), GU( n , q ), SL( n , q ), SU( n , q ), PGL( n , q ), PGU( n , q ), PSL( n , q ), PSL( n , q ), respectively. (See also ConjugacyClasses!attribute and Section Classical Groups.)

For each divisor f of n there is a group of Lie type with the same order as SL( n , q ), such that its derived subgroup modulo its center is isomorphic to PSL( n , q ). The various such groups with fixed n and q are called isogeneous. (Depending on congruence conditions on q and n several of these groups may actually be isomorphic.) The function NrConjugacyClassesSLIsogeneous computes the number of conjugacy classes in this group. The extreme cases f = 1 and f = n lead to the groups SL( n , q ) and PGL( n , q ), respectively.

The function NrConjugacyClassesSUIsogeneous is the analogous one for the corresponding unitary groups.

The formulae for the number of conjugacy classes are taken from Mac81.

gap> NrConjugacyClassesGL(24,27);
22528399544939174406067288580609952
gap> NrConjugacyClassesPSU(19,17);
15052300411163848367708
gap> NrConjugacyClasses(SL(16,16));
1229782938228219920

48.4 Constructors for Basic Groups

All 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 Cons, so for example CyclicGroup calls the constructor

The first argument cat 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 InstallMethod in ``Programming in GAP'' for the meaning of the arguments of InstallMethod).

InstallMethod( CyclicGroupCons,
    "regular perm group",
    true,
    [ IsPermGroup and IsRegularProp and IsFinite, IsInt and IsPosRat ], 0,
    function( filter, n )

    ...

    end );

48.5 Selection Functions

For a number of the group libraries two selection functions are provided. Each AllLibraryGroups selection function permits one to select all groups from the library Library that have a given set of properties. Currently, the library selection functions provided, of this type, are AllSmallGroups, AllIrreducibleSolvableGroups, AllTransitiveGroups, and AllPrimitiveGroups. Corresponding to each of these there is a OneLibraryGroup function (see OneLibraryGroup) which returns at most one group.

These 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 AllSmallGroups the first such function must be Size, and, unlike the other library selection functions, it supports an alternative syntax where Size is omitted (see AllSmallGroups). Also, see AllIrreducibleSolvableGroups, for details pertaining to AllIrreducibleSolvableGroups.

For an example, let us consider the selection function for the library of transitive groups (also see Transitive permutation groups). The command,

gap> AllTransitiveGroups(NrMovedPoints,[10..15],
                         Size,         [1..100],
                         IsAbelian,    false    );

Thus the AllTransitiveGroups behaves as if it was implemented by a function similar to the one defined below, where TransitiveGroupsList is a list of all transitive groups. (Note that in the definition below we assume for simplicity that AllTransitiveGroups 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.)

AllTransitiveGroups := function( fun1, val1, fun2, val2 )
local    groups, g, i;
  groups := [];
  for i  in [ 1 .. Length( TransitiveGroupsList ) ] do
    g := TransitiveGroupsList[i];
    if      fun1(g) = val1  or IsList(val1) and fun1(g) in val1
        and fun2(g) = val2  or IsList(val2) and fun2(g) in val2
     then
      Add( groups, g );
    fi;
  od;
  return groups;
end;

For each AllLibraryGroups function (see AllLibraryGroups) there is a corresponding function OneLibraryGroup on exactly the same arguments, i.e. there are OneSmallGroup, OneIrreducibleSolvableGroup, OneTransitiveGroup, and OnePrimitiveGroup. Each function simply returns the first group in the library (in the stored order) that has the prescribed properties, instead of all such groups. It returns fail if no such group exists in the library.

48.6 Transitive Permutation Groups

The transitive groups library currently contains representatives for all transitive permutation groups of degree at most 30. Two permutations groups of the same degree are considered to be equivalent, if there is a renumbering of points, which maps one group into the other one. In other words, if they lie in the same conjugacy class under operation of the full symmetric group by conjugation.

returns the nr-th transitive group of degree deg. Both deg and nr must be positive integers. The transitive groups of equal degree are sorted with respect to their size, so for example TransitiveGroup( deg, 1 ) is a transitive group of degree and size deg, e.g, the cyclic group of size deg, if deg is a prime.

returns the number of transitive groups of degree deg stored in the library of transitive groups. The function returns fail if deg is beyond the range of the library.

The selection functions (see Selection functions) for the transitive groups library are AllTransitiveGroups and OneTransitiveGroup. They obtain the following properties from the database without having to compute them anew:

NrMovedPoints, Size, Transitivity, and IsPrimitive.

This library was computed by Gregory Butler, John McKay, Gordon Royle and Alexander Hulpke. The list of transitive groups up to degree 11 was published in BM83, the list of degree 12 was published in Roy87, degree 14 and 15 were published in Butler93 and degrees 16--30 are dur to citeHulpke96,HulpkeTG. (Groups of prime degree of course are primitive and were known long before.)

The arrangement and the names of the groups of degree up to 15 is the same as given in ConwayHulpkeMcKay98. With the exception of the symmetric and alternating group (which are represented as SymmetricGroup and AlternatingGroup) the generators for these groups also conform to this paper with the only difference that 0 (which is not permitted in GAP for permutations to act on) is always replaced by the degree.

gap> TransitiveGroup(10,22);
S(5)[x]2
gap> l:=AllTransitiveGroups(NrMovedPoints,12,Size,1440,IsSolvable,false);
[ S(6)[x]2, M_10.2(12)=A_6.E_4(12)=[S_6[1/720]{M_10}S_6]2 ]
gap> List(l,IsSolvable);
[ false, false ]

Let G be a permutation group, acting transitively on a set of up to 30 points. Then TransitiveIdentification will return the position of this group in the transitive groups library. This means, if G acts on m points and TransitiveIdentification returns n, then G is permutation isomorphic to the group TransitiveGroup(m,n).

Note: The points moved do not need to be [1..n], the group á(2,3,4),(2,3)ñ is considered to be transitive on 3 points. If the group has several orbits on the points moved by it the result of TransitiveIdentification is undefined.

gap> TransitiveIdentification(Group((1,2),(1,2,3)));
2

48.7 Small Groups

The Small Groups library gives access to all groups of certain ``small'' orders. The groups are sorted by their orders and they are listed up to isomorphism; that is, for each of the available orders a complete and irredundant list of isomorphism type representatives of groups is given. Currently, the library contains the following groups:

• those of order at most 2000 except 1024   (423  164  062 groups);
• those of order 5⁵ and 7⁴   (92 groups);
• those of order qⁿ ·p where qⁿ divides 2⁸, 3⁶, 5⁵ or 7⁴ and p is an arbitrary prime with p ¹ q;
• those whose order factorises into at most 3 primes.

The library also has an identification function: it returns the library number of a given group. This function determines library numbers using invariants of groups. The function is available for all orders in the library except 512 and 1536.

The library is organised in 8 layers. Each layer contains the groups of certain orders and their corresponding group identification routines. It is possible to install the first n layers of the group library and the first m layers of the group identification for each 1 £ m £ n £ 8. This might be useful to save disk space. There is an extensive README file for the Small Groups library available in the small directory of the GAP distribution containing detailed information on the layers. In summary, the layers are:

1. the groups whose order factorises into at most 3 primes.
2. the remaining groups of order at most 1000 except 512 and 768.
3. the remaining groups of order 2ⁿ * p with n £ 8 and p an odd prime.
4. the groups of order 5⁵ and 7⁴ and the remaining groups of order qⁿ * p where qⁿ divides 3⁶, 5⁵ or 7⁴ and p is a prime with p ¹ q.
5. the remaining groups of order at most 2000 except 1024, 1152, 1536 and 1920.
6. the groups of orders 1152 and 1920.
7. the groups of order 512.
8. the groups of order 1536.

The data in this library has been carefully checked and cross-checked. It is believed to be reliable. However, no absolute guarantees are given and users should, as always, make their own checks in critical cases.

The data occupies about 26 MB (storing over 400 million groups in about 200 megabits). The group identification occupies about 42 MB of which 18 MB is used for the groups in layer (6). More information on the Small Groups library can be found on http://www.math.rwth-aachen.de/~Hans-Ulrich.Besche/small.html

This group library has been computed by Hans Ulrich Besche, Bettina Eick and E. A. O'Brien. A survey on this topic and an account of the history of group constructions can be found in BEO01. For further detailed information on the construction of this library we refer to New77, OBr90, OBr91, BescheEick98, BescheEick1000 and BescheEick768.

The Small Groups library incorporates the GAP 3 libraries TwoGroup and ThreeGroup. The data from these libraries was directly included into the Small Groups library, and the ordering there was preserved. The Small Groups library replaces the Gap 3 library of solvable groups of order at most 100. However, both the organisation and data descriptions of these groups has changed in the Small Groups library.

returns the i-th group of order order in the catalogue. If the group is solvable, it will be given as a PcGroup; otherwise it will be given as a permutation group. If the groups of order order are not installed, the function reports an error and enters a break loop.

returns all groups with certain properties as specified by arg. If arg is a number n, then this function returns all groups of order n. However, the function can also take several arguments which then must be organized in pairs function and value. In this case the first function must be Size and the first value an order or a range of orders. The function returns those groups of the specified orders having those properties specified by the remaining functions and their values.

Precomputed information is stored for the properties IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup, RankPGroup, PClassPGroup, LGLength, FrattinifactorSize and FrattinifactorId for the groups of order at most 2000 which have more than three prime factors, except those of order 512, 768, 1024, 1152, 1536, 1920 and those of order pⁿ ·q > 1000 with n > 2.

returns one group with certain properties as specified by arg. The permitted arguments are those supported by AllSmallGroups.

returns the number of groups of order order.

returns the library number of G; that is, the function returns a pair [order, i] where G is isomorphic to SmallGroup( order, i ).

similar to AllSmallGroups but returns ids instead of groups. This may prevent workspace overflows, if a large number of groups are expected in the output.

returns the catalogue number of G in the GAP 3 catalogue of solvable groups; that is, the function returns a pair [order, i] meaning that G is isomorphic to the group SolvableGroup( order, i ) in GAP 3.

prints information on the groups of the specified order.

GAP loads all necessary data from the library automatically, but it does not delete the data from the workspace again. Usually, this will be not necessary, since the data is stored in a compressed format. However, if a large number of groups from the library have been loaded, then the user might wish to remove the data from the workspace and this can be done by the above function call.

gap> G := SmallGroup( 768, 1000000 );
<pc group of size 768 with 9 generators>
gap> G := SmallGroup( [768, 1000000] );
<pc group of size 768 with 9 generators>

gap> AllSmallGroups( 6 );
[ <pc group of size 6 with 2 generators>, 
  <pc group of size 6 with 2 generators> ]
gap> AllSmallGroups( Size, 120, IsSolvableGroup, false );
[ Group([ (1,2,4,8)(3,6,9,5)(7,12,13,17)(10,14,11,15)(16,20,21,24)(18,22,19,
        23), (1,3,7)(2,5,10)(4,9,13)(6,11,8)(12,16,20)(14,18,22)(15,19,23)(17,
        21,24) ]), Group([ (1,2,3,4,5), (1,2) ]), 
  Group([ (1,2,3,5,4), (1,3)(2,4)(6,7) ]) ]

gap> G := OneSmallGroup( 120, IsNilpotentGroup, false );
<pc group of size 120 with 5 generators>
gap> IdSmallGroup(G);
[ 120, 1 ]
gap> G := OneSmallGroup( Size, [1..1000], IsSolvableGroup, false );
Group([ (1,2,3,4,5), (1,2,3) ])
gap> IdSmallGroup(G);
[ 60, 5 ]
gap> UnloadSmallGroupsData();

gap> IdSmallGroup( GL( 2,3 ) );
[ 48, 29 ]
gap> IdSmallGroup( Group( (1,2,3,4),(4,5) ) );
[ 120, 34 ]
gap> IdsOfAllSmallGroups( Size, 60, IsSupersolvableGroup, true );
[ [ 60, 1 ], [ 60, 2 ], [ 60, 3 ], [ 60, 4 ], [ 60, 6 ], [ 60, 7 ], 
  [ 60, 8 ], [ 60, 10 ], [ 60, 11 ], [ 60, 12 ], [ 60, 13 ] ]

gap> NumberSmallGroups( 512 );
10494213
gap> NumberSmallGroups( 2^8 * 23 );
1083472

gap> NumberSmallGroups( 2^9 * 23 );
Error, the library of groups of size 11776 is not available called from
<function>( <arguments> ) called from read-eval-loop
Entering break read-eval-print loop ...
you can 'quit;' to quit to outer loop, or
you can 'return;' to continue
brk> quit;
gap> 

gap> SmallGroupsInformation( 32 );

  There are 51 groups of order 32.
  They are sorted by their ranks. 
     1 is cyclic. 
     2 - 20 have rank 2.
     21 - 44 have rank 3.
     45 - 50 have rank 4.
     51 is elementary abelian. 

  For the selection functions the values of the following attributes 
  are precomputed and stored:
     IsAbelian, PClassPGroup, RankPGroup, FrattinifactorSize and 
     FrattinifactorId. 

  This size belongs to layer 2 of the SmallGroups library. 
  IdSmallGroup is available for this size. 
 

48.8 Finite Perfect Groups

The GAP library of finite perfect groups provides, up to isomorphism, a list of all perfect groups whose sizes are less than 10⁶ excluding the following sizes:

• For n = 61440, 122880, 172032, 245760, 344064, 491520, 688128, or 983040, the perfect groups of size n have not completely been determined yet. The library neither provides the number of these groups nor the groups themselves.
• For n = 86016, 368640, or 737280, the library does not yet contain the perfect groups of size n, it only provides their numbers which are 52, 46, and 54, respectively.

Except for these eleven sizes, the list of altogether 1096 perfect groups in the library is complete. It relies on results of Derek F. Holt and Wilhelm Plesken which are published in their book ``Perfect Groups'' HP89. Moreover, they have supplied us with files with presentations of 488 of the groups. In terms of these, the remaining 607 nontrivial groups in the library can be described as 276 direct products, 107 central products, and 224 subdirect products. They are computed automatically by suitable GAP functions whenever they are needed.

We are grateful to Derek Holt and Wilhelm Plesken for making their groups available to the GAP 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 GAP user who is interested in the groups.

The library has been brought into GAP format by Volkmar Felsch.

returns a group which is isomorphic to the library group specified by the size number [ size, n ] or by the two separate arguments size and n, assuming a default value of n = 1. The optional argument filt defines the filter in which the group is returned. Possible filters so far are IsPermGroup and IsSubgroupFpGroup. In the latter case, the generators and relators used coincide with those given in HP89.

gap> G := PerfectGroup(IsPermGroup,6048,1);
U3(3)

As 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.

gap> G:=PerfectGroup(IsPermGroup,129024,2);
L2(8) N ( 2^6 E 2^1 A ) C 2^1
gap> NrMovedPoints(G);
14336

This attribute is set for all groups obtained from the perfect groups library and has the value [size,nr] if the group is obtained with these parameters from the library.

returns the number of non-isomorphic perfect groups of size size for each positive integer size up to 10⁶ except for the eight sizes listed at the beginning of this section for which the number is not yet known. For these values as well as for any argument out of range it returns fail.

returns the number of perfect groups of size size which are available in the library of finite perfect groups. (The purpose of the function is to provide a simple way to formulate a loop over all library groups of a given size.)

SizeNumbersPerfectGroups 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 factor1, factor2, ... among their composition factors.

Each argument must either be the name of a 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.)

The following text strings are accepted as simple group names.

• An or A(n) for the alternating groups A_n, 5 £ n £ 9, for example A5 or A(6).
• Ln(q) or L(n,q) for PSL(n,q), where n Î {2,3} and q a prime power, ranging
  • for n=2 from 4 to 125
  • for n=3 from 2 to 5
• Un(q) or U(n,q) for PSU(n,q), where n Î {3,4} and q a prime power, ranging
  • for n=3 from 3 to 5
  • for n=4 from 2 to 2
• Sp4(4) or S(4,4) for the symplectic group S(4,4),
• Sz(8) for the Suzuki group Sz(8),
• Mn or M(n) for the Mathieu groups M₁₁, M₁₂, and M₂₂, and
• Jn or J(n) for the Janko groups J₁ and J₂.

Note 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 DisplayCompositionSeries command of GAP. However, as the names are compared as text strings, you are restricted to the above choice. Even expressions like L2(2^5) are not accepted.

As the use of the term PSU(n,q) is not unique in the literature, we mention that in this library it denotes the factor group of SU(n,q) by its centre, where SU(n,q) is the group of all n ×n unitary matrices with entries in GF(q²) and determinant 1.

The purpose of the function is to provide a simple way to formulate a loop over all library groups which contain certain composition factors.

DisplayInformationPerfectGroups displays some invariants of the n-th group of order size from the perfect groups library.

If no value of n has been specified, the invariants will be displayed for all groups of size size available in the library. The information provided for G includes the following items:

• a headline containing the size number [ size, n ] of G in the form size.n (the suffix .n will be suppressed if, up to isomorphism, G is the only perfect group of order size),
• a message if G is simple or quasisimple, i.e., if the factor group of G by its centre is simple,
• the ``description'' of the structure of G as it is given by Holt and Plesken in HP89 (see below),
• the size of the centre of G (suppressed, if G is simple),
• the prime decomposition of the size of G,
• orbit sizes for a faithful permutation representation of G which is provided by the library (see below),
• a reference to each occurrence of G in the tables of section 5.3 of HP89. Each of these references consists of a class number and an internal number (i,j) under which G 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.

gap> DisplayInformationPerfectGroups( 30720, 3 );
#I Perfect group 30720:  A5 ( 2^4 E N 2^1 E 2^4 ) A
#I   size = 2^11*3*5  orbit size = 240
#I   Holt-Plesken class 1 (9,3)
gap> DisplayInformationPerfectGroups( 30720, 6 );
#I Perfect group 30720:  A5 ( 2^4 x 2^4 ) C N 2^1
#I   centre = 2  size = 2^11*3*5  orbit size = 384
#I   Holt-Plesken class 1 (9,6)
gap> DisplayInformationPerfectGroups( Factorial( 8 ) / 2 );
#I Perfect group 20160.1:  A5 x L3(2) 2^1
#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 5 + 16
#I   Holt-Plesken class 31 (1,1) (occurs also in class 32)
#I Perfect group 20160.2:  A5 2^1 x L3(2)
#I   centre = 2  size = 2^6*3^2*5*7  orbit sizes = 7 + 24
#I   Holt-Plesken class 31 (1,2) (occurs also in class 32)
#I Perfect group 20160.3:  ( A5 x L3(2) ) 2^1
#I   centre = 2  size = 2^6*3^2*5*7  orbit size = 192
#I   Holt-Plesken class 31 (1,3)
#I Perfect group 20160.4:  simple group  A8
#I   size = 2^6*3^2*5*7  orbit size = 8
#I   Holt-Plesken class 26 (0,1)
#I Perfect group 20160.5:  simple group  L3(4)
#I   size = 2^6*3^2*5*7  orbit size = 21
#I   Holt-Plesken class 27 (0,1)

For any library group G, the library files do not only provide a presentation, but, in addition, a list of one or more subgroups S₁, ¼, S_r of G such that there is a faithful permutation representation of G of degree å_i=1^r [G:S_i] on the set { S_i g | 1 £ i £ r,  g Î G } of the cosets of the S_i. This allows one to construct the groups as permutation groups. The DisplayInformationPerfectGroups function displays only the available degree. The message

orbit size = 8

in the above example means that the available permutation representation is transitive and of degree 8, whereas the message

orbit sizes = 5 + 16

The notation used in the ``description'' of a group is explained in section 5.1.2 of HP89. We quote the respective page from there:

Within a class Q # p, an isomorphism type of groups will be denoted by an ordered pair of integers (r,n), where r ³ 0 and n > 0. More precisely, the isomorphism types in Q # p of order p^r |Q| will be denoted by (r,1), (r,2), (r,3), ¼ . Thus Q will always get the size number (0,1).

In addition to the symbol (r,n), the groups in Q # p 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 F_pQ-modules M with |Q|·|M| £ 10⁶, where F_p is the field of order p, are assigned symbols. These will either be simply p^x, where x 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 p^x, p^x¢, etc. The one-dimensional module with trivial Q-action will therefore be denoted by p¹. These symbols will be listed under the description of Q. 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 Q itself. (This convention is the most convenient in our context, but it is different from that adopted in the ATLAS CCN85, for example, where composition factors are listed in the reverse order. For example, we denote a group isomorphic to SL(2,5) by A₅ 2¹ rather than 2·A₅.)

Some 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.

×

between two factors denotes a direct product of F_pQ-modules or groups.

C

(for ``commutator'') between two factors means that the second lies in the commutator subgroup of the first. Similarly, a segment of the form (f₁ × f₂) C f₃ would mean that the factors f₁ and f₂ commute modulo f₃ and f₃ lies in [f₁,f₂].

A

(for ``abelian'') between two factors indicates that the second is in the pth power (but not the commutator subgroup) of the first. ``A'' may also follow the factors, if bracketed.

E

(for ``elementary abelian'') between two factors indicates that together they generate an elementary abelian group (modulo subsequent factors), but that the resulting F_pQ-module extension does not split.

N

(for ``nonsplit'') before a factor indicates that Q (or possibly its covering group) splits down as far at this factor but not over the factor itself. So ``Q f₁ N f₂'' means that the normal subgroup f₁f₂ of the group has no complement but, modulo f₂, f₁, does have a complement.

Brackets have their obvious meaning. Summarizing, we have:

×

= direct product;

C

= commutator subgroup;

A

= abelian;

E

= elementary abelian; and

N

= nonsplit.

Here are some examples.

1. A₅ (2⁴ E 2¹ E 2⁴) A means that the pairs 2⁴ E 2¹ and 2¹ E 2⁴ are both elementary abelian of exponent 4.

2. A₅ (2⁴ E 2¹ A) C 2¹ means that O₂(G) is of symplectic type 2¹⁺⁵, with Frattini factor group of type 2⁴ E 2¹. The ``A'' after the 2¹ indicates that G has a central cyclic subgroup 2¹ A 2¹ of order 4.

3. L₃(2) ((2¹ E) × (N 2³ E 2^3¢ A) C) 2^3¢ means that the 2^3¢ factor at the bottom lies in the commutator subgroup of the pair 2³ E 2^3¢ in the middle, but the lower pair 2^3¢ A 2^3¢ is abelian of exponent 4. There is also a submodule 2¹ E 2^3¢, and the covering group L₃(2) 2¹ of L₃(2) does not split over the 2³ factor. (Since G is perfect, it goes without saying that the extension L₃(2) 2¹ cannot split itself.)

We 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.

If such a group description has been given in the book for G (and, in fact, this is the case for most of the library groups), it is displayed by the DisplayInformationPerfectGroups function. Otherwise the function provides a less explicit description of the (in these cases unique) Holt-Plesken class to which G belongs, together with a serial number if this is necessary to make it unique.

48.9 Primitive Permutation Groups

GAP contains a library of primitive permutation groups which includes the following permutation groups up to permutation isomorphism (i.e., up to conjugacy in the corresponding symmetric group)

• the non-affine primitive permutation groups of degree £ 999, described in DixonMortimer88, with generators calculated in Theissen97,
• all primitive permutation groups of degree < 256, in particular,
  • the primitive permutation groups up to degree 50, calculated by C. Sims,
  • the solvable (hence affine) primitive permutation groups of degree < 256 as calculated by M. Short Sho92,
  • some insolvable affine primitive permutation groups of degree < 256 as calculated in Theissen97.

For degree up to 50, the names used are as given by Buekenhout and Leemans BuekenhoutLeemans96.

The names for the groups of higher degree are as chosen by Theissen97 and reflect the cohort structure in DixonMortimer88. They do not conform to the usual naming for extensions. That is l3,4.3 is the third (in some arbitrary ordering) group in a cohort for socle l3,4, but the socle factor is not necessarily of order 3.

The work in Theissen97 is known to have ommissions. Because of this we do not guarantee completeness of the lists beyond degree 50, though we have corrected errors as far as we know of them. When preparing the library it also has been ensured that the groups in it are all primitive and not conjugate.

In detail, we guarantee the following properties for this and further versions (but not versions which came before GAP 4.2) of the library:

• All groups in the library are primitive permutation groups of the indicated degree.
• The positions of the groups in the library are stable. That is PrimitiveGroup(n,nr) will always give you a permutation isomorphic group. Note however that we do not guarantee to keep the chosen S_n-representative, the generating set or the name for eternity.
• Different groups in the library are not conjugate in S_n.
• If a group in the library has a primitive subgroup with the same socle, this group is in the library as well.

(Note that the arrangement of groups is not guaranteed to be in increasing size, though it holds for many degrees.)

returns the primitive permutation group of degree deg with number nr from the list.

The arrangement of the groups differs from the arrangement of primitive groups in the list of C. Sims, which was used in GAP 3. See SimsNo (SimsNo).

returns the number of primitive permutation groups of degree deg in the library.

gap> NrPrimitiveGroups(25);
28
gap> PrimitiveGroup(25,19);
5^2:((Q_8:3)`4
gap> PrimitiveGroup(25,20);
ASL(2,5)
gap> PrimitiveGroup(25,22);
AGL(2,5)
gap> PrimitiveGroup(25,23);
(A_5xA_5):2

The selection functions (see Selection functions) for the primitive groups library are AllPrimitiveGroups and OnePrimitiveGroup. They obtain the following properties from the database without having to compute them anew:

NrMovedPoints, Size, Transitivity, ONanScottType, IsSimpleGroup, IsSolvableGroup, and SocleTypePrimitiveGroup.

(Note, that for groups of degree up to 999, O'Nan-Scott types 4a, 4b and 5 cannot occur.)

returns an iterator through AllPrimitiveGroups(attr1,val1,attr2,val2,...) without creating all these groups at the same time.

For a primitive permutation group for which an S_n-conjugate exists in the library of primitive permutation groups (see Primitive Permutation Groups), this attribute returns the index position. That is G is conjugate to PrimitiveGroup(NrMovedPoints(G),PrimitiveIdentification(G).

Methods only exist if the primitive groups library is installed.

Note: As this function uses the primitive groups library, the result is only guaranteed to the same extent as this library. If it is incomplete, PrimitiveIdentification might return an existing index number for a group not in the library.

gap> PrimitiveIdentification(Group((1,2),(1,2,3)));
2

If G is a primitive group obtained by PrimitiveGroup (respectively one of the selection functions) this attribute contains the number of the isomorphic group in the original list of C. Sims. (this is the arrangement as it was used in GAP 3.

gap> g:=PrimitiveGroup(25,2);
5^2:S_3
gap> SimsNo(g);
3

In DixonMortimer88 the primitive groups are sorted in ``cohorts'' according to their socle. For each degree, the variable COHORTS_PRIMITIVE_GROUPS contains a list of the cohorts for the primitive groups of this degree. Each cohort is represented by a list of length 2, the first entry specifies the socle type (see SocleTypePrimitiveGroup, section SocleTypePrimitiveGroup), the second entry listing the index numbers of the groups in this degree.

For example in degree 49, we have four cohorts with socles F₇², L₂(7)², A₇² and A₄₉ respectively. the group PrimitiveGroup(49,36), which is isomorphic to (A₇×A₇):2², lies in the third cohort with socle (A₇×A₇).

gap> COHORTS_PRIMITIVE_GROUPS[49];
[ [ rec( series := "Z", width := 2, parameter := 7 ),
      [ 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19,
          20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33 ] ],
  [ rec( series := "L", width := 2, parameter := [ 2, 7 ] ), [ 34 ] ],
  [ rec( series := "A", width := 2, parameter := 7 ), [ 35, 36, 37, 38 ] ],
  [ rec( series := "A", parameter := 49, width := 1 ), [ 39, 40 ] ] ]

48.10 Irreducible Solvable Matrix Groups

This function returns a representative of the i-th conjugacy class of irreducible solvable subgroup of GL(n,p), where n is an integer > 1, p is a prime, and p ⁿ < 256.

The numbering of the representatives should be considered arbitrary. However, it is guaranteed that the i-th group on this list will lie in the same conjugacy class in all future versions of GAP, unless two (or more) groups on the list are discovered to be duplicates, in which case IrreducibleSolvableMatrixGroup will return fail for all but one of the duplicates.

For values of n, p, and i admissible to IrreducibleSolvableGroup, IrreducibleSolvableMatrixGroup returns a representative of the same conjugacy class of subgroups of GL(n,p) as IrreducibleSolvableGroup. Note that it currently adds two more groups (missing from the original list by Mark Short) for n = 2, p = 13.

This function returns the number of conjugacy classes of irreducible solvable subgroup of GL(n,p).

This function returns a list of conjugacy class representatives G of matrix groups over a prime field such that func_i (G) = val_i or func_i (G) Î val_i . The following possibilities for func_i are particularly efficient, because the values can be read off the information in the data base: DegreeOfMatrixGroup (or Dimension or DimensionOfMatrixGroup) for the linear degree, Characteristic for the field characteristic, Size, IsPrimitiveMatrixGroup (or IsLinearlyPrimitive), and MinimalBlockDimension.

This function returns one solvable subgroup G of a matrix group over a prime field such that func_i (G) = val_i or func_i (G) Î val_i for all i. The following possibilities for func_i are particularly efficient, because the values can be read off the information in the data base: DegreeOfMatrixGroup (or Dimension or DimensionOfMatrixGroup) for the linear degree, Characteristic for the field characteristic, Size, IsPrimitiveMatrixGroup (or IsLinearlyPrimitive), and MinimalBlockDimension.

This variable provides a way to get from irreducible solvable groups to primitive groups and vice versa. For the group G=IrreducibleSolvableGroup( n, p, k ) and d=pⁿ, the entry PrimitiveIndexIrreducibleSolvableGroup[d][i] gives the index number of the semidirect product pⁿ:G in the library of primitive groups.

Searching for an index Position in this list gives the translation in the other direction.

This function is obsolete, because for n = 2, p = 13, two groups were missing from the underlying database. It has been replaced by the function IrreducibleSolvableGroupMS (see IrreducibleSolvableGroupMS). Please note that the latter function does not guarantee any ordering of the groups in the database. However, for values of n, p, and i admissible to IrreducibleSolvableGroup, IrreducibleSolvableGroupMS returns a representative of the same conjugacy class of subgroups of GL(n,p) as IrreducibleSolvableGroup did before.

48.11 Irreducible Maximal Finite Integral Matrix Groups

A library of irreducible maximal finite integral matrix groups is provided with GAP. It contains Q-class representatives for all of these groups of dimension at most 31, and Z-class representatives for those of dimension at most 11 or of dimension 13, 17, 19, or 23.

The 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 Z-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 Z-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 Q-class representatives of the groups of non-prime dimensions between 12 and 24 and the groups of dimensions 25 to 31 (see PN95, NP95, Neb95, Neb96).

The library has been brought into GAP format by Volkmar Felsch. He has applied several GAP 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 GAP.

In the preceding acknowledgement, we used some notations that will also be needed in the sequel. We first define these.

Any integral matrix group G of dimension n is a subgroup of GL_n(Z) as well as of GL_n(Q) and hence lies in some conjugacy class of integral matrix groups under GL_n(Z) and also in some conjugacy class of rational matrix groups under GL_n(Q). As usual, we call these classes the Z-class and the Q-class of G, respectively. Note that any conjugacy class of subgroups of GL_n(Q) contains at least one Z-class of subgroups of GL_n(Z) and hence can be considered as the Q-class of some integral matrix group.

In the context of this library we are only concerned with Z-classes and Q-classes of subgroups of GL_n(Z) which are irreducible and maximal finite in GL_n(Z) (we will call them i.m.f. subgroups of GL_n(Z)). We can distinguish two types of these groups:

First, there are those i.m.f. subgroups of GL_n(Z) which are also maximal finite subgroups of GL_n(Q). Let us denote the set of their Q-classes by Q₁(n). It is clear from the above remark that Q₁(n) just consists of the Q-classes of i.m.f. subgroups of GL_n(Q).

Secondly, there is the set Q₂(n) of the Q-classes of the remaining i.m.f. subgroups of GL_n(Z), i.e., of those which are not maximal finite subgroups of GL_n(Q). For any such group G, say, there is at least one class C Î Q₁(n) such that G is conjugate under Q to a proper subgroup of some group H Î C. In fact, the class C is uniquely determined for any group G occurring in the library (though there seems to be no reason to assume that this property should hold in general). Hence we may call C the rational i.m.f. class of G. Finally, we will denote the number of classes in Q₁(n) and Q₂(n) by q₁(n) and q₂(n), respectively.

As an example, let us consider the case n = 4. There are 6 Z-classes of i.m.f. subgroups of GL₄(Z) with representative subgroups G₁, ¼, G₆ of isomorphism types G₁ @ W(F₄), G₂ @ D₁₂ wr C₂, G₃ @ G₄ @ C₂ ×S₅, G₅ @ W(B₄), and G₆ @ (D₁₂ Y D₁₂) : C₂. The corresponding Q-classes, R₁, ¼, R₆, say, are pairwise different except that R₃ coincides with R₄. The groups G₁, G₂, and G₃ are i.m.f. subgroups of GL₄(Q), but G₅ and G₆ are not because they are conjugate under GL₄(Q) to proper subgroups of G₁ and G₂, respectively. So we have Q₁(4) = { R₁, R₂, R₃ }, Q₂(4) = { R₅, R₆ }, q₁(4) = 3, and q₂(4) = 2.

The q₁(n) Q-classes of i.m.f. subgroups of GL_n(Q) have been determined for each dimension n £ 31. The current GAP library provides integral representative groups for all these classes. Moreover, all Z-classes of i.m.f. subgroups of GL_n(Z) are known for n £ 11 and for n Î {13,17,19,23}. For these dimensions, the library offers integral representative groups for all Q-classes in Q₁(n) and Q₂(n) as well as for all Z-classes of i.m.f. subgroups of GL_n(Z).

Any group G of dimension n given in the library is represented as the automorphism group G = Aut(F,L) = { g Î GL_n(Z) | Lg = L   and   g F g^tr = F } of a positive definite symmetric n ×n matrix F Î Z^{n ×n} on an n-dimensional lattice L @ Z^{1 ×n} (for details see e.g. PN95). GAP provides for G a list of matrix generators and the Gram matrix F.

The positive definite quadratic form defined by F defines a norm v F v^tr for each vector v Î L, and there is only a finite set of vectors of minimal norm. These vectors are often simply called the short vectors. Their set splits into orbits under G, and G being irreducible acts faithfully on each of these orbits by multiplication from the right. GAP provides for each of these orbits the orbit size and a representative vector.

Like most of the other GAP libraries, the library of i.m.f. integral matrix groups supplies an extraction function, ImfMatrixGroup. 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, ImfInvariants and DisplayImfInvariants, which provide some Z-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.

We 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 irreducible maximal finite integral matrix groups would have to include this term in full length and hence would grow extremely long. Therefore we have decided to use the abbreviation Imf instead in order to restrict the names to some reasonable length.

The first three functions can be used to formulate loops over the classes.

ImfNumberQQClasses returns the number q₁(dim) of Q-classes of i.m.f. rational matrix groups of dimension dim. Valid values of dim are all positive integers up to 31.

Note: In order to enable you to loop just over the classes belonging to Q₁(dim), we have arranged the list of Q-classes of dimension dim for any dimension dim in the library such that, whenever the classes of Q₂(dim) are known, too, i.e., in the cases dim £ 11 or dim Î {13,17,19,23}, the classes of Q₁(dim) precede those of Q₂(dim) and hence are numbered from 1 to q₁(dim).

ImfNumberQClasses returns the number of Q-classes of groups of dimension dim which are available in the library. If dim £ 11 or dim Î {13,17,19,23}, this is the number q₁(dim) + q₂(dim) of Q-classes of i.m.f. subgroups of GL_dim(Z). Otherwise, it is just the number q₁(dim) of Q-classes of i.m.f. subgroups of GL_dim(Q). Valid values of dim are all positive integers up to 31.

ImfNumberZClasses returns the number of Z-classes in the q^th Q-class of i.m.f. integral matrix groups of dimension dim. Valid values of dim are all positive integers up to 11 and all primes up to 23.

DisplayImfInvariants displays the following Z-class invariants of the groups in the z^th Z-class in the q^th Q-class of i.m.f. integral matrix groups of dimension dim:

• its Z-class number in the form dim.q.z, if dim is at most 11 or a prime at most 23, or its Q-class number in the form dim.q, else,
• a message if the group is solvable,
• the size of the group,
• the isomorphism type of the group,
• the elementary divisors of the associated quadratic form,
• the sizes of the orbits of short vectors (these sizes are the degrees of the faithful permutation representations which you may construct using the functions IsomorphismPermGroup or IsomorphismPermGroupImfGroup below),
• the norm of the associated short vectors,
• only in case that the group is not an i.m.f. group in GL_n(Q): an appropriate message, including the Q-class number of the corresponding rational i.m.f. class.

If you specify the value 0 for any of the parameters dim, q, or z, the command will loop over all available dimensions, Q-classes of given dimension, or Z-classes within the given Q-class, respectively. Otherwise, the values of the arguments must be in range. A value z ¹ 1 must not be specified if the Z-classes are not known for the given dimension, i.e., if dim > 11 and dim Ï {13,17,19,23}. The default value of z is 1. This value of z will be accepted even if the Z-classes are not known. Then it specifies the only representative group which is available for the q^th Q-class. The greatest legal value of dim is 31.

gap> DisplayImfInvariants( 3, 1, 0 );
#I Z-class 3.1.1:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = W(B3)
#I   elementary divisors = 1^3
#I   orbit size = 6, minimal norm = 1
#I Z-class 3.1.2:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I   elementary divisors = 1*4^2
#I   orbit size = 8, minimal norm = 3
#I Z-class 3.1.3:  Solvable, size = 2^4*3
#I   isomorphism type = C2 wr S3 = C2 x S4 = C2 x W(A3)
#I   elementary divisors = 1^2*4
#I   orbit size = 12, minimal norm = 2
gap> DisplayImfInvariants( 8, 15, 1 );
#I Z-class 8.15.1:  Solvable, size = 2^5*3^4
#I   isomorphism type = C2 x (S3 wr S3)
#I   elementary divisors = 1*3^3*9^3*27
#I   orbit size = 54, minimal norm = 8
#I   not maximal finite in GL(8,Q), rational imf class is 8.5
gap> DisplayImfInvariants( 20, 23 );
#I Q-class 20.23:  Size = 2^5*3^2*5*11
#I   isomorphism type = (PSL(2,11) x D12).C2
#I   elementary divisors = 1^18*11^2
#I   orbit size = 3*660 + 2*1980 + 2640 + 3960, minimal norm = 4

Note that the function DisplayImfInvariants uses a kind of shorthand to display the elementary divisors. E. g., the expression 1*3^3*9^3*27 in the preceding example stands for the elementary divisors 1,3,3,3,9,9,9,27. (See also the next example which shows that the function ImfInvariants provides the elementary divisors in form of an ordinary GAP list.)

In the description of the isomorphism types the following notations are used:

A x B

denotes a direct product of a group A by a group B,

A subd B

denotes a subdirect product of A by B,

A Y B

denotes a central product of A by B,

A wr B

denotes a wreath product of A by B,

A:B

denotes a split extension of A by B,

A·B

denotes just an extension of A by B (split or nonsplit).

The groups involved are

• the cyclic groups C_n, dihedral groups D_n, and generalized quaternion groups Q_n of order n, denoted by Cn, Dn, and Qn, respectively,
• the alternating groups A_n and symmetric groups S_n of degree n, denoted by An and Sn, respectively,
• the linear groups GL_n(q), PGL_n(q), SL_n(q), and PSL_n(q), denoted by GL(n,q), PGL(n,q), SL(n,q), and PSL(n,q), respectively,
• the unitary groups SU_n(q) and PSU_n(q), denoted by SU(n,q) and PSU(n,q), respectively,
• the symplectic groups Sp(n,q) and PSp(n,q), denoted by Sp(n,q) and PSp(n,q), respectively,
• the orthogonal groups O₈ ⁺(2) and PO₈ ⁺(2), denoted by O+(8,2) and PO+(8,2), respectively,
• the extraspecial groups 2₊ ¹⁺⁸, 3₊ ¹⁺², 3₊ ¹⁺⁴, and 5₊ ¹⁺², denoted by 2+^(1+8), 3+^(1+2), 3+^(1+4), and 5+^(1+2), respectively,
• the Chevalley group G₂(3), denoted by G2(3),
• the twisted Chevalley group ³D₄(2), denoted by 3D4(2),
• the Suzuki group Sz(8), denoted by Sz(8),
• the Weyl groups W(A_n), W(B_n), W(D_n), W(E_n), and W(F₄), denoted by W(An), W(Bn), W(Dn), W(En), and W(F4), respectively,
• the sporadic simple groups Co₁, Co₂, Co₃, HS, J₂, M₁₂, M₂₂, M₂₃, M₂₄, and Mc, denoted by Co1, Co2, Co3, HS, J2, M12, M22, M23, M24, and Mc, respectively,
• a point stabilizer of index 11 in M₁₁, denoted by M10.

As mentioned above, the data assembled by the function DisplayImfInvariants are ``cheap data'' 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.

ImfInvariants returns a record which provides some Z-class invariants of the groups in the z^th Z-class in the q^th Q-class of i.m.f. integral matrix groups of dimension dim. A value z ¹ 1 must not be specified if the Z-classes are not known for the given dimension, i.e., if dim > 11 and dim Ï {13,17,19,23}. The default value of z is 1. This value of z will be accepted even if the Z-classes are not known. Then it specifies the only representative group which is available for the q^th Q-class. The greatest legal value of dim is 31.

The resulting record contains six or seven components:

size

the size of any representative group G,

isSolvable

is true if G is solvable,

isomorphismType

a text string describing the isomorphism type of G (in the same notation as used by the function DisplayImfInvariants above),

elementaryDivisors

the elementary divisors of the associated Gram matrix F (in the same format as the result of the function ElementaryDivisorsMat, see ElementaryDivisorsMat),

minimalNorm

the norm of the associated short vectors,

sizesOrbitsShortVectors

the sizes of the orbits of short vectors under F,

maximalQClass

the Q-class number of an i.m.f. group in GL_n(Q) that contains G as a subgroup (only in case that not G itself is an i.m.f. subgroup of GL_n(Q)).

Note 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 Z-class invariants, but also Q-class invariants.

Note further that, though the isomorphism type is a Q-class invariant, you will sometimes get different descriptions for different Z-classes of the same Q-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.

gap> ImfInvariants( 8, 15, 1 );
rec( size := 2592, isSolvable := true, isomorphismType := "C2 x (S3 wr S3)", 
  elementaryDivisors := [ 1, 3, 3, 3, 9, 9, 9, 27 ], minimalNorm := 8, 
  sizesOrbitsShortVectors := [ 54 ], maximalQClass := 5 )
gap> ImfInvariants( 24, 1 ).size;
10409396852733332453861621760000
gap> ImfInvariants( 23, 5, 2 ).sizesOrbitsShortVectors;
[ 552, 53130 ]
gap> for i in [ 1 .. ImfNumberQClasses( 22 ) ] do
>    Print( ImfInvariants( 22, i ).isomorphismType, "\n" ); od;
C2 wr S22 = W(B22)
(C2 x PSU(6,2)).S3
(C2 x S3) wr S11 = (C2 x W(A2)) wr S11
(C2 x S12) wr C2 = (C2 x W(A11)) wr C2
C2 x S3 x S12 = C2 x W(A2) x W(A11)
(C2 x HS).C2
(C2 x Mc).C2
C2 x S23 = C2 x W(A22)
C2 x PSL(2,23)
C2 x PSL(2,23)
C2 x PGL(2,23)
C2 x PGL(2,23)

ImfMatrixGroup is the essential extraction function of this library (note that its name has been changed from ImfMatGroup in GAP 3 to ImfMatrixGroup in GAP 4). It returns a representative group, G say, of the z^th Z-class in the q^th Q-class of i.m.f. integral matrix groups of dimension dim. A value z ¹ 1 must not be specified if the Z-classes are not known for the given dimension, i.e., if dim > 11 and dim Ï {13,17,19,23}. The default value of z is 1. This value of z will be accepted even if the Z-classes are not known. Then it specifies the only representative group which is available for the q^th Q-class. The greatest legal value of dim is 31.

gap> G := ImfMatrixGroup( 5, 1, 3 );
ImfMatrixGroup(5,1,3)
gap> for m in GeneratorsOfGroup( G ) do PrintArray( m ); od;
[ [  -1,   0,   0,   0,   0 ],
  [   0,   1,   0,   0,   0 ],
  [   0,   0,   0,   1,   0 ],
  [  -1,  -1,  -1,  -1,   2 ],
  [  -1,   0,   0,   0,   1 ] ]
[ [  0,  1,  0,  0,  0 ],
  [  0,  0,  1,  0,  0 ],
  [  0,  0,  0,  1,  0 ],
  [  1,  0,  0,  0,  0 ],
  [  0,  0,  0,  0,  1 ] ]

The attributes Size and IsSolvable will be properly set in the resulting matrix group G. In addition, it has two attributes IsImfMatrixGroup and ImfRecord where the first one is just a logical flag set to true 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 ImfInvariants described above (see ImfInvariants), namely the components isomorphismType, elementaryDivisors, minimalNorm, and sizesOrbitsShortVectors and, if G is not a rational i.m.f. group, maximalQClass. Moreover, it has the two components

form

the associated Gram matrix F, and

repsOrbitsShortVectors

representatives of the orbits of short vectors under F.

The last one of these components will be required by the function IsomorphismPermGroup below.

Example:

gap> Size( G );
3840
gap> imf := ImfRecord( G );;
gap> imf.isomorphismType;
"C2 wr S5 = C2 x W(D5)"
gap> PrintArray( imf.form );
[ [  4,  0,  0,  0,  2 ],
  [  0,  4,  0,  0,  2 ],
  [  0,  0,  4,  0,  2 ],
  [  0,  0,  0,  4,  2 ],
  [  2,  2,  2,  2,  5 ] ]
gap> imf.elementaryDivisors;
[ 1, 4, 4, 4, 4 ]
gap> imf.minimalNorm;
4

If you want to perform calculations in such a matrix group G you should be aware of the fact that the permutation group routines of GAP 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 G on one of the orbits of the associated short vectors. You may call one of the following functions IsomorphismPermGroup or IsomorphismPermGroupImfGroup to get an isomorphism to such a permutation group (note that these GAP 4 functions have replaced the GAP 3 functions PermGroup and PermGroupImfGroup).

returns an isomorphism, j say, from the given i.m.f. integral matrix group G to a permutation group P : = j(G ) acting on a minimal orbit, S say, of short vectors of G such that each matrix m Î G is mapped to the permutation induced by its action on S.

Note that in case of a large orbit the construction of j may be space and time consuming. Fortunately, there are only six Q-classes in the library for which the smallest orbit of short vectors is of size greater than 20000, the worst case being the orbit of size 196560 for the Leech lattice (dim = 24, q = 3).

The inverse isomorphism j^-1 from P to G is constructed by determining a Q-base B Ì S of Q^{1 ×dim} in S and, in addition, the associated base change matrix M which transforms B into the standard base of Z^{1 ×dim}. This allows a simple computation of the preimage j^-1(p) of any permutation p Î P as follows. If, for 1 £ i £ dim, b_i is the position number in S of the i^th base vector in B, it suffices to look up the vector whose position number in S is the image of b_i under p and to multiply this vector by M to get the i^th row of j^-1(p).

You may use the functions Image and PreImage (see Image and PreImage) to switch from G to P and back from P to G.

As an example, let us continue the preceding example and compute the solvable residuum of the group G.

gap> # Perform the computations in an isomorphic permutation group.
gap> phi := IsomorphismPermGroup( G );;
gap> P := Image( phi );
Group([ (1,7,6)(2,9)(4,5,10), (2,3,4,5)(6,9,8,7) ])
gap> D := DerivedSubgroup( P );
Group([ (1,2,10,9)(3,8)(4,5)(6,7), (1,3,10,8)(2,5)(4,7)(6,9), 
  (1,2,8,7,5)(3,4,6,10,9) ])
gap> Size( D );
960
gap> IsPerfectGroup( D );
true
gap> # We have found the solvable residuum of P,
gap> # now move the results back to the matrix group G.
gap> R := PreImage( phi, D );
<matrix group of size 960 with 3 generators>
gap> for m in GeneratorsOfGroup( R ) do PrintArray( m ); od;
[ [  -1,  -1,  -1,  -1,   2 ],
  [   0,  -1,   0,   0,   0 ],
  [   0,   0,   0,   1,   0 ],
  [   0,   0,   1,   0,   0 ],
  [  -1,  -1,   0,   0,   1 ] ]
[ [   0,   0,   0,   1,   0 ],
  [  -1,  -1,  -1,  -1,   2 ],
  [   0,   0,  -1,   0,   0 ],
  [   1,   0,   0,   0,   0 ],
  [   0,  -1,  -1,   0,   1 ] ]
[ [   0,  -1,   0,   0,   0 ],
  [   0,   0,   1,   0,   0 ],
  [   0,   0,   0,  -1,   0 ],
  [   1,   1,   1,   1,  -2 ],
  [   0,   0,   1,   0,  -1 ] ]

IsomorphismPermGroupImfGroup returns an isomorphism, j say, from the given i.m.f. integral matrix group G to a permutation group P acting on the n ^th orbit, S say, of short vectors of G such that each matrix m Î G is mapped to the permutation induced by its action on S.

The only difference to the above function IsomorphismPermGroup 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 IsomorphismPermGroup( G ) has been implemented such that it is equivalent to IsomorphismPermGroupImfGroup( G, 1 ).

gap> ImfInvariants( 12, 9 ).sizesOrbitsShortVectors;
[ 120, 300 ]
gap> G := ImfMatrixGroup( 12, 9 );
ImfMatrixGroup(12,9)
gap> phi1 := IsomorphismPermGroupImfGroup( G, 1 );;
gap> P1 := Image( phi1 );
<permutation group of size 2400 with 2 generators>
gap> LargestMovedPoint( P1 );
120
gap> phi2 := IsomorphismPermGroupImfGroup( G, 2 );;
gap> P2 := Image( phi2 );
<permutation group of size 2400 with 2 generators>
gap> LargestMovedPoint( P2 );
300

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GAP 4 manual
May 2002