# Semidirect product of Z3 and D8 with action kernel V4

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## Definition

This group is defined by the following presentation:

$G := \langle a_1, a_2,a_3,a_4 \mid a_1^3 = a_2^2 = a_3^2 = a_4^2 = e, a_1a_2 = a_2a_1, a_1a_3 = a_3a_1, a_2a_3 = a_3a_2, a_4a_1a_4^{-1} = a_1^{-1}, a_4a_2a_4^{-1} = a_3, a_4a_3a_4^{-1} = a_3^{-1} \rangle$

## 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 $G$:

gap> G := SmallGroup(24,8);

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

IdGroup(G) = [24,8]

or just do:

IdGroup(G)

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

### Description by presentation

gap> F := FreeGroup(4);
<free group on the generators [ f1, f2, f3, f4 ]>
gap> G := F/[F.1^3,F.2^2,F.3^2,F.1*F.2*F.1^(-1)*F.2^(-1), F.1*F.3*F.1^(-1)*F.3^(-1), F.2*F.3*F.2^(-1)*F.3^(-1), F.4^2,F.4*F.1*F.4^(-1)*F.1,F.4*F.2*F.4^(-1)*F.3];