# Nontrivial semidirect product of cyclic groups of prime-square order

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

This group can be defined as the semidirect product of a cyclic group of prime-square order and a cyclic group of prime-square order acting nontrivially on it.

Explicitly, it is given by:

$G := \langle x,y \mid x^{p^2} = y^{p^2} = e, yxy^{-1} = x^{p + 1} \rangle$.

Its GAP ID is $(p^4,4)$ for those primes $p$ where these GAP IDs are defined.

Particular cases are nontrivial semidirect product of Z4 and Z4 for $p = 2$ and nontrivial semidirect product of Z9 and Z9 for $p = 3$.

### Alternative definitions

The group can be defined in the following alternative ways: