Direct product of Mp^3 and Zp

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 satisfies property::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.