# Holomorph of Z8

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## Contents

## Definition

This group (which we shall call throughout) can be defined in either of these ways:

- It is the holomorph of the cyclic group on eight elements. In other words, it is the semidirect product of the cyclic group on eight elements, with its automorphism group.
- It is the holomorph of the ring . In other words, it is the general affine group .

The group has the following presentation (with denoting the identity element):

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 32#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 32 and has ID 43 among the groups of order 32 in GAP's SmallGroup library. For context, there are 51 groups of order 32. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(32,43)`

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

`gap> G := SmallGroup(32,43);`

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

`IdGroup(G) = [32,43]`

or just do:

`IdGroup(G)`

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

### Other definitions

The group can be defined using GAP's AutomorphismGroup and SemidirectProduct functions. Here is a full code snippet:

gap> C := CyclicGroup(8); <pc group of size 8 with 3 generators> gap> SemidirectProduct(AutomorphismGroup(C),C); <pc group with 5 generators>

This can be compressed by coding a function Holomorph for computing the holomorph of a group. With this function coded, we can use:

Holomorph(CyclicGroup(8))