# Nontrivial semidirect product of Z4 and Z8

## Definition

This group is defined by the presentation: $\! G := \langle a,b \mid a^4 = b^8 = e, bab^{-1} = a^3 \rangle$

## GAP implementation

### Group ID

This finite group has order 32 and has ID 12 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(32,12)

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

gap> G := SmallGroup(32,12);

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

IdGroup(G) = [32,12]

or just do:

IdGroup(G)

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

### Description by presentation

The group can be described in GAP using its presentation:

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^4,F.2^8,F.2*F.1*F.2^(-1)*F.1];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 32, 12 ]