Faithful semidirect product of cyclic p-groups

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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.