# Semidirect product of Z16 and Z4 via fifth power map

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## Definition

This group is defined as the external semidirect product with base normal subgroup cyclic group:Z16 and acting quotient group cyclic group:Z4, where the generator of the latter acts as the fifth power map.

Equivalently, it is given by the following presentation: $G := \langle a,b \mid a^{16} = b^4 = e, bab^{-1} = a^5 \rangle$

Here, $e$ denotes the identity element.

## GAP implementation

### Group ID

This finite group has order 64 and has ID 28 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(64,28)

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

gap> G := SmallGroup(64,28);

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

IdGroup(G) = [64,28]

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(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^(-5)];
<fp group on the generators [ f1, f2 ]>
gap> IdGroup(G);
[ 64, 28 ]