Nontrivial semidirect product of cyclic groups of prime-square order

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.