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.
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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:

• For odd $p$ it it is the second omega subgroup of the Sylow subgroup of holomorph of cyclic group of prime-cube order.