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 $p$ 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.