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

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 $$M_{p^4}$$.

GAP implementation
This satisfies property::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.