# Semidirect product of Z3 and D8 with action kernel V4

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## Contents

## Definition

This group is defined as follow: it is the external semidirect product of cyclic group:Z3 by dihedral group:D8 where the action of the latter on the former is given by a homomorphism whose kernel is one of the Klein four-subgroups of dihedral group:D8. Note that the homomorphism is completely specified by its kernel because there is a unique isomorphism between any two groups isomorphic to cyclic group:Z2.

## Arithmetic functions

Want to compare and contrast arithmetic function values with other groups of the same order? Check out groups of order 24#Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 24 and has ID 8 among the groups of order 24 in GAP's SmallGroup library. For context, there are 15 groups of order 24. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(24,8)`

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

`gap> G := SmallGroup(24,8);`

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

`IdGroup(G) = [24,8]`

or just do:

`IdGroup(G)`

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