There are two definitions of a metacyclic group.
1. A metacyclic group is a group 
such that both its commutator
subgroup 
and the quotient group 
are cyclic (Rose 1994,
p. 247).
2. A group 
is metacyclic if it has a cyclic normal
subgroup 
such that the quotient group 
is also cyclic (Rose 1994, p. 56).
In general, a group may be metacyclic according to the second definition and fail the first one. For example, the quaternion group 
has a normal cyclic subgroup of order
4, thus it satisfies definition (2). On the other hand, the commutant 
consists of two elements 
, the quotient 
is isomorphic to the finite
group C2×C2, and thus the group is not cyclic.
The first definition is more classical, but nowadays essentially all algebraists use the second definition, which is the one used in the remainder of this article.
Metacyclic groups are solvable and have a composition series of length two.
A complete classification of finite metacyclic groups has been given by Hempel (2000). The metacyclic groups are all generated by two elements which are subject to three
relations depending on several numerical parameters. As a special, consider the groups
generated by two elements 
and 
such that
 |
where 
.
For 
and 
this is the definition of the dihedral group 
, where 
is the rotation of 
around the center and 
is the reflection around the axis of symmetry of a regular
-gon.
Dihedral groups are metacyclic, as is every subgroup and every quotient group of a metacyclic group (Rose 1994, p. 56).
A group is always metacyclic if its Sylow p-subgroups are all cyclic (Scott 1987, p. 356; Rose 1994, pp. 246-247). In particular, it follows that every group of squarefree order is metacyclic.
