# Groups of order 2^m.3^n

This article discusses groups whose order has prime factors in the set and no others. Another way of putting this is that the article discusses finite -groups where .

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.

## Contents

## Orders of interest

The clickable links in the table are to pages devoted to groups of that order.

The rows represent values of and corresponding values of . The columns represent values of and corresponding values of .

generic | . | |||||||||
---|---|---|---|---|---|---|---|---|---|---|

-- | -- | |||||||||

generic | generic | 2^m | 3.2^m | 3^2.2^m | 3^3.2^m | 3^4.2^m | 3^5.2^m | 3^6.2^m | 3^7.2^m | |

0 | 1 | 3^n | 1 | 3 | 9 | 27 | 81 | 243 | 729 | 2187 |

1 | 2 | 2.3^n | 2 | 6 | 18 | 54 | 162 | 486 | 1458 | 4374 |

2 | 4 | 2^2.3^n | 4 | 12 | 36 | 108 | 324 | 972 | 2916 | 8748 |

3 | 8 | 2^3.3^n | 8 | 24 | 72 | 216 | 648 | 1944 | 5832 | |

4 | 16 | 2^4.3^n | 16 | 48 | 144 | 432 | 1296 | 3888 | ||

5 | 32 | 2^5.3^n | 32 | 96 | 288 | 864 | ||||

6 | 64 | 2^6.3^n | 64 | 192 | 576 | 1728 | ||||

7 | 128 | 2^7.3^n | 128 | 384 | 1152 | |||||

8 | 256 | 2^8.3^n | 256 | 768 | ||||||

9 | 512 | 2^9.3^n | 512 | 1536 | ||||||

10 | 1024 | 2^10.3^n | 1024 |

## Counts of groups

### Counts of all groups

The rows give values of and the columns give values of . Each cell value is the total number of isomorphism classes of groups of order .

Note that for small values of , the order of magnitude of the values depends roughly on (so it is roughly of the same order of magnitude as we move along a north-east-to-south-west diagonal). This fails to hold for larger values of .

. | |||||||||
---|---|---|---|---|---|---|---|---|---|

-- | -- | ||||||||

0 | 1 | 1 | 1 | 2 | 5 | 15 | 67 | 504 | 9310 |

1 | 2 | 1 | 2 | 5 | 15 | 55 | 261 | 1798 | |

2 | 4 | 2 | 5 | 14 | 45 | 176 | 900 | ||

3 | 8 | 5 | 15 | 50 | 177 | 757 | 3973 | ||

4 | 16 | 14 | 52 | 197 | 775 | 3609 | |||

5 | 32 | 51 | 231 | 1045 | 4725 | ||||

6 | 64 | 267 | 1543 | 8681 | 47937 | ||||

7 | 128 | 2328 | 20169 | 157877 | |||||

8 | 256 | 56092 | 1090235 | ||||||

9 | 512 | 10494213 | 408641062 | ||||||

10 | 1024 | 49487365422 |

### Counts of nilpotent groups

See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.

Thus, the number of nilpotent groups of order = (number of groups of order ) (number of groups of order )

. | |||||||||
---|---|---|---|---|---|---|---|---|---|

-- | -- | ||||||||

0 | 1 | 1 | 1 | 2 | 5 | 15 | 67 | 504 | 9310 |

1 | 2 | 1 | 1 | 2 | 5 | 15 | 67 | 504 | 9310 |

2 | 4 | 2 | 2 | 4 | 10 | 30 | 134 | 1008 | 18620 |

3 | 8 | 5 | 5 | 10 | 25 | 75 | 335 | 2520 | 46550 |

4 | 16 | 14 | 14 | 28 | 70 | 210 | 938 | 7056 | 130340 |

5 | 32 | 51 | 51 | 102 | 255 | 765 | 3417 | 25704 | 474810 |

6 | 64 | 267 | 267 | 534 | 1335 | 4005 | 17889 | 134568 | 2485770 |

7 | 128 | 2328 | |||||||

8 | 256 | 56092 | |||||||

9 | 512 | 10494213 | |||||||

10 | 1024 | 49487365422 |