# Inner automorphism group of wreath product of Z5 and Z5

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## Contents

## Definition

This group is the inner automorphism group of wreath product of groups of order p for the prime . Equivalently, it is the inner automorphism group of the wreath product of Z5 and Z5.

## Arithmetic functions

## GAP implementation

### Group ID

This finite group has order 3125 and has ID 30 among the groups of order 3125 in GAP's SmallGroup library. For context, there are 77 groups of order 3125. It can thus be defined using GAP's SmallGroup function as:

`SmallGroup(3125,30)`

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

`gap> G := SmallGroup(3125,30);`

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

`IdGroup(G) = [3125,30]`

or just do:

`IdGroup(G)`

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