# Direct product of Mp^3 and Zp

This article is about a family of groups with a parameter that is prime. For any fixed value of the prime, we get a particular group.

View other such prime-parametrized groups

## Definition

Let be an odd prime number. This group is defined as the external direct product of the group (defined as the semidirect product of cyclic group of prime-square order and cyclic group of prime order) and (the group of prime order).

## GAP implementation

This finite group has order the fourth power of the prime, i.e., . For odd , it has ID 13 for among the groups of order in GAP's SmallGroup library. For , it has ID 11 among the groups of order 16. For context, there are 15 groups of order for odd and 14 groups of order for .

For odd , it can thus be defined using GAP's SmallGroup function as follows, assuming is specified beforehand:

`SmallGroup(p^4,13)`

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

`gap> G := SmallGroup(p^4,13);`

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

`IdGroup(G) = [p^4,13]`

or just do:

`IdGroup(G)`

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