# Holomorphs

Given a group $G$ and the full group of automorphisms $A$ of $G$ then the holomorph of $G$ is the semidirect product $G \times_\theta A$, where $\theta: A \rightarrow Aut(G)$ 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 $G \rightarrow E$, and the natural epimorphism $E \rightarrow A$ 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 $G \rightarrow E$, and the natural epimorphism $E \rightarrow A$ are also returned. In the returned group $E$, the generators of $G$ appear first, followed by those of $A$.

## `Example: holomorph (ex-fba5ab)`

We construct the holomorph of the group $G = PGL(2, 9)$.

```magma
> 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

```