# Groups of order 18

From Groupprops

This article gives information about, and links to more details on, groups of order 18

See pages on algebraic structures of order 18| See pages on groups of a particular order

## The list

Group | GAP ID |
---|---|

dihedral group:D18 | (18,1) |

cyclic group:Z18 | (18,2) |

direct product of S3 and Z3 | (18,3) |

generalized dihedral group for E9 | (18,4) |

direct product of Z6 and Z3 | (18,5) |

## Arithmetic functions

Function | dihedral group:D18 | cyclic group:Z18 | direct product of S3 and Z3 | generalized dihedral group for E9 | direct product of Z6 and Z3 |
---|---|---|---|---|---|

exponent | 18 | 18 | 6 | 6 | 6 |

derived length | 2 | 1 | 2 | 2 | 1 |

nilpotency class | -- | 1 | -- | -- | 1 |

Fitting length | 2 | 1 | 2 | 2 | 1 |

## Group properties

Property | dihedral group:D18 | cyclic group:Z18 | direct product of S3 and Z3 | generalized dihedral group for E9 | direct product of Z6 and Z3 |
---|---|---|---|---|---|

cyclic group | No | Yes | No | No | No |

abelian group | No | Yes | No | No | Yes |

nilpotent group | No | Yes | No | No | Yes |

metacyclic group | Yes | Yes | Yes | No | Yes |

supersolvable group | Yes | Yes | Yes | No | Yes |

solvable group | Yes | Yes | Yes | Yes | Yes |