Nontrivial semidirect product of cyclic groups of prime-square order

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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.
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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: