# Semidirect product of Z16 and Z4 of M-type

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## Definition

This group is defined as the external semidirect product of cyclic group:Z16 by cyclic group:Z4 acting by the $9^{th}$ power map. Explicitly, it is given by the presentation:

$G := \langle a,b \mid a^{16} = b^4 = e, bab^{-1} = a^9 \rangle$

## GAP implementation

### Group ID

This finite group has order 64 and has ID 27 among the groups of order 64 in GAP's SmallGroup library. For context, there are groups of order 64. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(64,27)

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

gap> G := SmallGroup(64,27);

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

IdGroup(G) = [64,27]

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 defined using a presentation as follows:

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^(16),F.2^4,F.2*F.1*F.2^(-1)*F.1^(-9)];
<fp group on the generators [ f1, f2 ]>