# Groups of order 3^2.2^n

From Groupprops

This article discusses the groups of order , where varies over nonnegative integers. Note that any such group has a 3-Sylow subgroup of order , and a 2-Sylow subgroup, which is of order . Further, because order has only two prime factors implies solvable, any such group is a solvable group and in particular a finite solvable group.

## Number of groups of small orders

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

0 | 1 | 9 | 2 | See groups of order 9, classification of groups of prime-square order |

1 | 2 | 18 | 5 | See groups of order 18 |

2 | 4 | 36 | 14 | See groups of order 36 |

3 | 8 | 72 | 50 | See groups of order 72 |

4 | 16 | 144 | 197 | See groups of order 144 |

5 | 32 | 288 | 1045 | See groups of order 288 |

6 | 64 | 576 | 8681 | See groups of order 576 |

7 | 128 | 1152 | 157877 | See groups of order 1152 |

## Arithmetic functions

### Derived length

There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

total number of groups | length 1 (abelian groups) |
length 2 | length 3 | length 4 | length 5 | ||
---|---|---|---|---|---|---|---|

0 | 9 | 2 | 2 | ||||

1 | 18 | 5 | 2 | 3
| |||

2 | 36 | 14 | 4 | 10
| |||

3 | 72 | 50 | 6 | 37 |
7 | ||

4 | 144 | 197 | 10 | 159 |
22 | 6 | |

5 | 288 | 1045 | 14 | 876 |
123 | 32 | |

6 | 576 | 8681 | 22 | 7625 |
800 | 234 | |

7 | 1152 | 157877 | 30 | ? |