Central product of UT(3,p) and Zp^2

Definition
Let $$p$$ be a prime number. This group is defined as the central product of unitriangular matrix group:UT(3,p) and the cyclic group of prime-square order $$\mathbb{Z}/p^2\mathbb{Z}$$ sharing a common central subgroup that is cyclic of order $$p$$.

GAP implementation
This satisfies property::finite group has order the fourth power of the prime, i.e., $$p^4$$. It has ID 14 among the groups of order $$p^4$$ in GAP's SmallGroup library for odd $$p$$, and ID 13 among groups of order $$p^4$$ for $$p = 2$$. 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 odd and specified beforehand:

SmallGroup(p^4,14)

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

gap> G := SmallGroup(p^4,14);

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^4,14]

or just do:

IdGroup(G)

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