# Faithful semidirect product of cyclic p-groups

## Definition

This is a general type of group of prime power order obtained as follows. Consider natural numbers $l < k$, and an odd prime $p$. Now, the multiplicative group of $\mathbb{Z}/p^k\mathbb{Z}$ contains a cyclic subgroup of order $p^l$: the subgroup generated multiplicatively by $p^{k-l} + 1$.

The group we are interested in is the semidirect product of $\mathbb{Z}/p^k\mathbb{Z}$ 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 $k,l$.