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

From Groupprops

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:

.

Its GAP ID is for those primes where these GAP IDs are defined.

Particular cases are nontrivial semidirect product of Z4 and Z4 for and nontrivial semidirect product of Z9 and Z9 for .

### Alternative definitions

The group can be defined in the following alternative ways:

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