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

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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.