# 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 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 sharing a common central subgroup that is cyclic of order .

## GAP implementation

This finite group has order the fourth power of the prime, i.e., . It has ID 14 among the groups of order in GAP's SmallGroup library for odd , and ID 13 among groups of order for . For context, there are 15 groups of order for odd and 14 groups of order for . It can thus be defined using GAP's SmallGroup function as follows, assuming 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 :

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

Conversely, to check whether a given group 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.