MAGMA Computational Algebra System

Holomorphs

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(G) : Grp -> GrpPerm, HomGrp, HomGrp

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(G, A) : Grp, GrpAuto -> GrpPerm, HomGrp, HomGrp

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.

Example GrpAuto_holomorph (H63E9)

> 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