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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: 20000
SmallOuterAutGroup := 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: 0
The level of verbose printing. The possible values are 0, 1, 2 or 3.
PrintSearchCount: RngIntElt Default: 1000
PrintSearchCount := 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.
We create a non-soluble group G of 4 x 4 matrices defined over
the field of 8-th roots of unity and construct its automorphism group.
> 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.
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