# Holomorph of D8

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## Definition

This group is defined in the following eqiuvalent ways:

- It is the holomorph of the dihedral group of order eight, i.e., the semidirect product of the dihedral group of order eight and its automorphism group (which is also isomorphic to the dihedral group of order eight).
- It is the -Sylow subgroup of the holomorph of the quaternion group.

## GAP implementation

### Group ID

This finite group has order 64 and has ID 134 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(64,134)`

For instance, we can use the following assignment in GAP to create the group and name it :

`gap> G := SmallGroup(64,134);`

Conversely, to check whether a given group is in fact the group we want, we can use GAP's IdGroup function:

`IdGroup(G) = [64,134]`

or just do:

`IdGroup(G)`

to have GAP output the group ID, that we can then compare to what we want.

### Alternative descriptions

The group can be constructed using the GAP commands DihedralGroup, AutomorphismGroup, and SemidirectProduct:

gap> G := DihedralGroup(8); <pc group of size 8 with 3 generators> gap> A := AutomorphismGroup(G); <group of size 8 with 3 generators> gap> S := SemidirectProduct(A,G); <pc group with 6 generators>

is the group we need.

It can also be constructed using a hand-coded GAP function: Holomorph, with which it becomes:

`Holomorph(DihedralGroup(8))`