# Nontrivial semidirect product of Z3 and Z8

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## Definition

This group is defined as the external semidirect product of cyclic group:Z3 by cyclic group:Z8, where the generator of the latter acts on the former via the inverse map. Explicitly, it is given by: $\langle a,x \mid a^3 = x^8 = e, xax^{-1} = a^{-1} \rangle$

where $e$ denotes the identity element.

## GAP implementation

### Group ID

This finite group has order 24 and has ID 1 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(24,1)

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

gap> G := SmallGroup(24,1);

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

IdGroup(G) = [24,1]

or just do:

IdGroup(G)

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

### Description by presentation

Here is a description using the presentation given in the definition:

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^3,F.2^8,F.2*F.1*F.2^(-1)*F.1];