Groups of order 3^n

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	1396077	unknown	see groups of order 6561
9	19683	5937876645	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	1396077	?	?	?
9	19683	5937876645	?	?	?

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]

We use the function SortArithmeticFunctionSizes, which is not in-built but is easy to code (follow link to get code). We also use the in-built function NilpotencyClassOfGroup. Using these, we get:

gap> SortArithmeticFunctionSizes(3,0,NilpotencyClassOfGroup);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(3,1,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(3,2,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(3,3,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(3,4,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 6 ], [ 3, 4 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(3,5,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 28 ], [ 3, 26 ], [ 4, 6 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(3,6,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 133 ], [ 3, 282 ], [ 4, 71 ], [ 5, 7 ], [ 6, 0 ] ]
gap> SortArithmeticFunctionSizes(3,7,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 15 ], [ 2, 1757 ], [ 3, 6050 ], [ 4, 1309 ], [ 5, 173 ], [ 6, 6 ], [ 7, 0 ] ]

Note that for the $3^{7}$ case, the new version of the SmallGroup library (obtain here) is needed, and additional memory may need to be allocated using the command line option `-o' (e.g., gap -o 5G).

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