# Semidirect product of Z5 and Z8 via square map

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## Definition

This group of order 40 is defined by means of the following presentation (here, $e$ is used to denote the identity element): $G := \langle a,b \mid a^5 = b^8 = e, bab^{-1} = a^2 \rangle$

## GAP implementation

### Group ID

This finite group has order 40 and has ID 3 among the groups of order 40 in GAP's SmallGroup library. For context, there are groups of order 40. It can thus be defined using GAP's SmallGroup function as:

SmallGroup(40,3)

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

gap> G := SmallGroup(40,3);

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

IdGroup(G) = [40,3]

or just do:

IdGroup(G)

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

### Description by presentation

We can use the presentation to define the group in GAP:

gap> F := FreeGroup(2);
<free group on the generators [ f1, f2 ]>
gap> G := F/[F.1^5,F.2^8,F.2*F.1*F.2^(-1)*F.1^(-2)];
<fp group on the generators [ f1, f2 ]>

To verify that this is the correct group:

gap> IdGroup(G);
[ 40, 3 ]