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

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

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:

SmallGroup(p^4,6)

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:

IdGroup(G)

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