# Metacyclic group

From Groupprops

This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism

View a complete list of group propertiesVIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## Contents

## Definition

### Symbol-free definition

A **metacyclic group** is a group having a cyclic normal subgroup with a cyclic quotient group.

### Definition with symbols

A group is termed **metacyclic** if there exists a normal subgroup of such that both and are cyclic.

## Relation with other properties

### Stronger properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

Cyclic group |

### Weaker properties

Property | Meaning | Proof of implication | Proof of strictness (reverse implication failure) | Intermediate notions |
---|---|---|---|---|

supersolvable group | ||||

virtually abelian group | metacyclic implies virtually abelian | |||

polycyclic group | ||||

metabelian group | ||||

solvable group |