# Characteristically metacyclic not implies metacyclic derived series

From Groupprops

This article gives the statement and possibly, proof, of a non-implication relation between two group properties. That is, it states that every group satisfying the first group property (i.e., characteristically metacyclic group) neednotsatisfy the second group property (i.e., group with metacyclic derived series)

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## Statement

It is possible to have a group with a Cyclic characteristic subgroup (?) such that is also cyclic, but such that the abelianization is *not* cyclic.

## Proof

### Example of the dihedral group

`Further information: dihedral group:D8`

Consider the dihedral group of order eight:

.

Let be the four-element cyclic subgroup . Then, both and are cyclic, but the abelianization is isomorphic to a Klein four-group, which is not cyclic.