# Metacyclic group

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This article defines a group property: a property that can be evaluated to true/false for any given group, invariant under isomorphism
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VIEW RELATED: Group property implications | Group property non-implications |Group metaproperty satisfactions | Group metaproperty dissatisfactions | Group property satisfactions | Group property dissatisfactions

## 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 $G$ is termed metacyclic if there exists a normal subgroup $N$ of $G$ such that both $N$ and $G/N$ 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