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

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 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 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.