# Faithful semidirect product of cyclic p-groups

From Groupprops

## Definition

This is a general type of group of prime power order obtained as follows. Consider natural numbers , and an odd prime . Now, the multiplicative group of contains a cyclic subgroup of order : the subgroup generated multiplicatively by .

The group we are interested in is the semidirect product of with this cyclic group.

## Group properties

### Solvable length

The group is a semidirect product of one cyclic group by another, so it is a metacyclic group. In particular, it is a metabelian group: it has solvable length two.

### Nilpotence class

The class of this group depends on .