Semidirect product of cyclic group of prime-cube order and cyclic group of prime order

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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


This group is defined as the external semidirect product of the cyclic group of order p^3 with the unique subgroup of order p in its automorphism group. The group has order p^4 and is sometimes denoted M_{p^4}.

Particular cases

Value of prime p Value p^4 Corresponding group M_{p^4}
2 16 M16
3 81 M81

GAP implementation

This finite group has order the fourth power of the prime, i.e., p^4, and has ID 6 among the groups of order p^4 in GAP's SmallGroup library. For context, there are 15 groups of order p^4 for odd p and 14 groups of order p^4 for p = 2. It can thus be defined using GAP's SmallGroup function as follows, assuming p is specified beforehand:


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

gap> G := SmallGroup(p^4,6);

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^4,6]

or just do:


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