An automorphism group of the finite group G may be created in one of two ways. Firstly, the full automorphism group of G may be constructed by invoking an appropriate lifting algorithm. Secondly, an arbitrary group of automorphisms A of G may be created by giving a set of generators for A defined in terms of their action on a set of generators for G.
Given a finite group G, construct the full automorphism group F of G. The group G may be a permutation group, a (finite) matrix group or a finite soluble group given by a pc-presentation. The function returns the full automorphism group of G as a group of mappings (i.e., as a group of type GrpAuto). If G is a permutation or matrix group, then the automorphism group F is also computed as a finitely presented group and can be accessed via the function FPGroup(F). A function PermutationRepresentation is provided that when applied to F attempts to construct a faithful permutation representation of reasonable degree (see below).SmallOuterAutGroup: RngIntElt Default: 20000SmallOuterAutGroup := t: Specify the strategy for the backtrack search when testing an automorphism for lifting to the next layer. If the outer automorphism group O at the previous level has order at most t, then the regular representation of O is used, otherwise the program tries to find a smaller degree permutation representation of O.Print: RngIntElt Default: 0The level of verbose printing. The possible values are 0, 1, 2 or 3.PrintSearchCount: RngIntElt Default: 1000PrintSearchCount := s: If Print := 3, then a message is printed at each s-th iteration during the backtrack search for lifting automorphisms.In the case of a non-soluble group, the algorithm described in Cannon and Holt [CH03] is used. If G is a p-group of type GrpPC the algorithm described in Eick, Leedham-Green and O'Brien [ELGO02] is used. For more details see Section p-group. If G is of type GrpPC but is not a p-group, the algorithm of Smith [Smi94], as extended by Smith and Slattery, is used. For more details see Section Automorphism Group.
When G is a non-soluble permutation or matrix group, the algorithm relies on a database of automorphism groups for the non-cyclic simple factors of G, hence the non-abelian composition factors of G must belong to a restricted list. In V2.11 this list includes all simple groups of order at most 1.6times107, the alternating groups of degree at most 1000, all groups from several generic families, including PSL(2, q), PSL(3, q), PSL(4, p), PSL(5, p), PSU(3, p) and PSp(4, p) and the sporadic groups M11, M12, M22, M23, M24, J1, J2, J3, HS, McL, Co3, He and others. The list is being extended regularly.
> L<zeta_8> := CyclotomicField(8);
> i := -zeta_8^2;
> t := zeta_8^3;
> G := MatrixGroup< 4, L |
> [ 1/2, 1/2, 1/2, 1/2,
> 1/2,-1/2, 1/2,-1/2,
> 1/2, 1/2,-1/2,-1/2,
> 1/2,-1/2,-1/2, 1/2 ],
> DiagonalMatrix( [1,1,1,-1] ),
> DiagonalMatrix( [1,i,1,i] ),
> DiagonalMatrix( [t,t,t,t] ) >;
> Order(G);
92160
> CompositionFactors(G);
G
| Cyclic(2)
*
| Alternating(6)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
1
> A := AutomorphismGroup(G);
> Order(A);
92160
Let G be a finite group and let Q be a sequence of elements which generate G. Let φ1, ..., φr be a sequence of automorphisms of G that generate the group of automorphisms A. The group A is specified by a sequence I of length r where the i-th term of I defines φi in terms of a sequence containing the images of the elements of Q under the action of φi. The function returns the group of automorphisms A of G.