Direct product of Mp^3 and Zp

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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 p be an odd prime number. This group is defined as the external direct product of the group M_{p^3} (defined as the semidirect product of cyclic group of prime-square order and cyclic group of prime order) and \mathbb{Z}_p (the group of prime order).

GAP implementation

This finite group has order the fourth power of the prime, i.e., p^4. For odd p, it has ID 13 for among the groups of order p^4 in GAP's SmallGroup library. For p = 2, it has ID 11 among the groups of order 16. For context, there are 15 groups of order p^4 for odd p and 14 groups of order p^4 for p = 2.

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

SmallGroup(p^4,13)

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

gap> G := SmallGroup(p^4,13);

Conversely, to check whether a given group G 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.