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Given a group G and the full group of automorphisms A of G then
the holomorph of G is the semidirect product G x θ A,
where θ: A -> Aut(G) is the identity map.
Holomorph(GrpFP, G) : Grp -> Cat[GrpFP], GrpFP, HomGrp, HomGrp
Given a finite permutation, matrix or pc-group G with full group of
automorphisms A, this function returns the semidirect product E of
G by A. The group E is returned as a permutation group (or a
finitely presented group if GrpFP is specified) of degree
|G| in which G is a regular normal subgroup, and A is the stabilizer
of the point 1. The embedding map G -> E, and the natural
epimorphism E -> A are also returned. In the returned group
E, the generators of G appear first, followed by those of A.
Holomorph(GrpFP, G, A) : Cat[GrpFP], Grp, GrpAuto -> GrpFP, HomGrp, HomGrp
Given a finite permutation, matrix or pc-group G and a group of
automorphisms A, this function returns the semidirect product E of
G by A. The group E is returned as a permutation group (or a
finitely presented group if GrpFP is specified) of degree
|G| in which G is a regular normal subgroup, and A is the stabilizer
of the point 1. The embedding map G -> E, and the natural
epimorphism E -> A are also returned. In the returned group
E, the generators of G appear first, followed by those of A.
We construct the holomorph of the group G = PGL(2, 9).
> G := PGL(2, 9);
> E := Holomorph(G); E;
Permutation group E acting on a set of cardinality 720
> #E;
1036800
> CompositionFactors(E);
G
| Cyclic(2)
*
| Cyclic(2)
*
| Cyclic(2)
*
| Alternating(6)
*
| Alternating(6)
1
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