Characteristically metacylic group

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Definition

A group $G$ is termed characteristically metacyclic if there exists a characteristic subgroup $H$ of $G$ such that both $H$ and the quotient group (note that characteristic implies normal, so the quotient group does exist) $G/H$ are cyclic groups.

Relation with other properties

Weaker properties

Property	Meaning	Proof of implication	Proof of strictness (reverse implication failure)	Intermediate notions
metacyclic group
characteristically polycyclic group				|FULL LIST, MORE INFO
polycyclic group		(via metacyclic)	(via metacyclic)	|FULL LIST, MORE INFO
supersolvable group		(via metacyclic)	(via metacyclic)	|FULL LIST, MORE INFO
solvable group		(via metacyclic)	(via metacyclic)	|FULL LIST, MORE INFO