The term "metacyclic group" has two definitions.
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 cyclic normal
subgroup of order 4, thus it satisfies definition
(2). On the other hand, the commutator subgroup 
consists of two elements 
, the quotient
group 
is isomorphic to the finite
group C2×C2, and thus the quotient
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, since their commutator subgroups are cyclic.
A complete classification of finite metacyclic groups has been given by Hempel (2000). These groups are all generated
by two elements which are subject to three relations
depending on several numerical parameters. As a special case, 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
n-gon.
Dihedral groups are metacyclic, as is every subgroup and every quotient group of a metacyclic group (Rose 1994, p. 56).
A finite group whose Sylow p-subgroups are all cyclic is metacyclic
even in the first sense. Both its commutator subgroup 
and the quotient
group 
are cyclic (Scott 1987, p. 356; Rose 1994, pp. 246-247).
In particular, it follows that every group of squarefree order is metacyclic.
