# Groups of order 3^n

## Contents

## Number of groups of small orders

FACTS TO CHECK AGAINST(number of groups of prime power order):Exact counts: Higman-Sims asymptotic formula on number of groups of prime power order |Upper and lower bounds: Higman-Sims asymptotic formula on number of groups of prime power order (best known) | upper bound on number of groups of prime power order using power-commutator presentations (very crude) | inductive upper bound on number of groups of prime power order using power-commutator presentations (very crude)

Exponent | Value | Number of groups of order | Greatest integer function of logarithm of number of groups to base 3 | Reason/Explanation/List |
---|---|---|---|---|

0 | 1 | 1 | 0 | only trivial group |

1 | 3 | 1 | 0 | only cyclic group:Z3; see equivalence of definitions of group of prime order |

2 | 9 | 2 | 0 | cyclic group:Z9 and elementary abelian group:E9; see groups of order 9; see also classification of groups of prime-square order |

3 | 27 | 5 | 1 | see groups of order 27; see also classification of groups of prime-cube order |

4 | 81 | 15 | 2 | see groups of order 81; see also classification of groups of prime-fourth order for odd prime |

5 | 243 | 67 | 3 | see groups of order 243 |

6 | 729 | 504 | 5 | see groups of order 729 |

7 | 2187 | 9310 | 8 | see groups of order 2187 |

8 | 6561 | unknown | unknown | see groups of order 6561 |

9 | 19683 | unknown | unknown | see groups of order 19683 |

## Counts for various equivalence classes

### Isoclinism

Exponent | Value | Number of groups of order | Number of equivalence classes under isoclinism | Sizes of equivalence classes (i.e., number of groups in each equivalence class) | Additional note |
---|---|---|---|---|---|

0 | 1 | 1 | 1 | 1 | trivial group only |

1 | 3 | 1 | 1 | 1 | cyclic group:Z3 only |

2 | 9 | 2 | 1 | 2 | both groups are abelian groups |

3 | 27 | 5 | 2 | 3,2 | abelian groups form one equivalence class, non-abelian groups form another. See groups of order 27#Families and classification or groups of prime-cube order#Families and classification. |

4 | 81 | 15 | 3 | 5,6,4 | One equivalence class for each nilpotency class value. Class one (abelian groups), class two, and class three (maximal class groups). See groups of order 81#Families and classification or groups of prime-fourth order#Families and classification |

5 | 243 | 67 | ? | ? | ? |

6 | 729 | 504 | ? | ? | ? |

7 | 2187 | 9310 | ? | ? | ? |

8 | 6561 | ? | ? | ? | ? |

9 | 19683 | ? | ? | ? | ? |

## Arithmetic functions

`Further information: arithmetic functions for groups of order 3^n`

Below is the data for nilpotency class. Data for other arithmetic functions is available at arithmetic functions for groups of order 3^n.

total number of groups | class 0 | class 1 | class 2 | class 3 | class 4 | class 5 | class 6 | ||
---|---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 1 | ||||||

1 | 3 | 1 | 0 | 1
| |||||

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

3 | 27 | 5 | 0 | 3 |
2 | ||||

4 | 81 | 15 | 0 | 5 | 6 |
4 | |||

5 | 243 | 67 | 0 | 7 | 28 |
26 | 6 | ||

6 | 729 | 504 | 0 | 11 | 133 | 282 |
71 | 7 | |

7 | 2187 | 9310 | 0 | 15 | 1757 | 6050 |
1309 | 173 | 6 |

Here is the same information, now given in terms of the fraction of groups of a given order that are of a given nilpotency class. For ease of comparison, all fractions are written as decimals, rounded to the fourth decimal place.

total number of groups | average of values (equal weighting on all groups) | class 0 | class 1 | class 2 | class 3 | class 4 | class 5 | class 6 | ||
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 0 | 1 | ||||||

1 | 3 | 1 | 1 | 0 | 1 | |||||

2 | 9 | 2 | 1 | 0 | 1 | |||||

3 | 27 | 5 | 1.4 | 0 | 0.6000 |
0.4000 | ||||

4 | 81 | 15 | 1.9333 | 0 | 0.3333 | 0.4000 |
0.2667 | |||

5 | 243 | 67 | 2.4627 | 0 | 0.1045 | 0.4179 |
0.3881 | 0.1343 | ||

6 | 729 | 504 | 2.8611 | 0 | 0.0218 | 0.2639 | 0.5595 |
0.1409 | 0.1389 | |

7 | 2187 | 9310 | 2.9876 | 0 | 0.0016 | 0.1887 | 0.6498 |
0.1406 | 0.0186 | 0.0006 |

Below is information for the probability distribution of nilpotency class using the cohomology tree probability distribution:

total number of groups | average of values (cohomology tree probability distribution) | class 0 | class 1 | class 2 | class 3 | class 4 | class 5 | class 6 | ||
---|---|---|---|---|---|---|---|---|---|---|

0 | 1 | 1 | 0 | 1
| ||||||

1 | 3 | 3 | 1 | 0 | 1
| |||||

2 | 9 | 2 | 1 | 0 | 1
| |||||

3 | 27 | 5 | 1.2222 | 0 | 0.7778 |
0.2222 | ||||

4 | 81 | 15 | 1.4701 | 0 | 0.5519 |
0.4262 | 0.0219 | |||

5 | 243 | 67 | 1.8282 | 0 | 0.3778 | 0.4177 |
0.2028 | 0.0016 |

## Elements

`Further information: element structure of groups of order 3^n`

## Subgroups

`Further information: subgroup structure of groups of order 3^n`

## Linear representation theory

`Further information: linear representation theory of groups of order 3^n`