# Groups of order 3^n

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## 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 $n$ Value $3^n$ Number of groups of order $3^n$ 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 $n$ Value $3^n$ Number of groups of order $3^n$ 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. $n$ $3^n$ 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 GAP code to generate this information: [SHOW MORE]

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. $n$ $3^n$ 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: $n$ $3^n$ 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