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


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:



gap> G := SmallGroup(p^4,13);


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

or just do:

