# Groups of order 2.3^n

From Groupprops

This article discusses the groups of order , where varies over nonnegative integers. Note that any such group has a *normal* 3-Sylow subgroup and a complement to that which is a 2-Sylow subgroup isomorphic to cyclic group:Z2. Thus, the group is either a nilpotent group (a direct product of the 3-Sylow subgroup and cyclic group:Z2) or can be described as the internal semidirect product of a group of order by a group of order two whose non-identity element acts as a non-identity automorphism.

See also groups of order 3^n.

## Number of groups of small orders

Exponent | Value | Value | Number of groups of order | Reason/explanation/list |
---|---|---|---|---|

0 | 1 | 2 | 1 | cyclic group:Z2, see also equivalence of definitions of group of prime order |

1 | 3 | 6 | 2 | See groups of order 6. The groups are cyclic group:Z6 and symmetric group:S3. |

2 | 9 | 18 | 5 | See groups of order 18. |

3 | 27 | 54 | 15 | See groups of order 54. |

4 | 81 | 162 | 55 | See groups of order 162. |

5 | 243 | 486 | 261 | See groups of order 486. |

6 | 729 | 1458 | 1798 | See groups of order 1458. |