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 .