Groups of order 7^n

Number of groups of small orders

Exponent $n$	Value $57^{n}$	Number of groups of order $7^{n}$	Reason/Explanation/List
1	7	1	only cyclic group:Z7; see equivalence of definitions of group of prime order
2	49	2	cyclic group:Z49 and elementary abelian group:E49; see also groups of order 49 and classification of groups of prime-square order
3	343	5	see groups of order 343 and classification of groups of prime-cube order
4	2401	15	see groups of order 2401 and classification of groups of prime-fourth order for odd prime
5	16807	83	see groups of order 16807 and also the PORC formula in the table in the next section.
6	117649	860	see groups of order 117649 and also the PORC formula in the table in the next section.
7	823543	113147	see groups of order 823453

Arithmetic functions

Nilpotency class

$n$	$7^{n}$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1	1
1	7	1	0	1
2	49	2	0	2
3	343	5	0	3	2
4	2401	15	0	5	6	4
5	16807	83	0	7	32	33	11
6	117649	860	0	11	165	508	133	43

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 functions, the above data can be generated as follows:

gap> SortArithmeticFunctionSizes(7,0,NilpotencyClassOfGroup);
[ [ 0, 1 ] ]
gap> SortArithmeticFunctionSizes(7,1,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(7,2,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(7,3,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(7,4,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 6 ], [ 3, 4 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(7,5,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 32 ], [ 3, 33 ], [ 4, 11 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(7,6,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 165 ], [ 3, 508 ], [ 4, 133 ], [ 5, 43 ], [ 6, 0 ] ]

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.

$n$	$7^{n}$	total number of groups	class 0	class 1	class 2	class 3	class 4	class 5	class 6
0	1	1	1
1	7	1	0	1
2	49	2	0	1
3	343	5	0	0.6000	0.4000
4	2401	15	0	0.3333	0.4000	0.2667
5	16807	83	0	0.0843	0.3855	0.3976	0.1325
6	117649	860	0	0.0128	0.1919	0.5907	0.1547	0.0500

Derived length

$n$	$7^{n}$	total number of groups	length 0	length 1	length 2	length 3
0	1	1	1
1	7	1	0	1
2	49	2	0	2
3	343	5	0	3	2
4	2401	15	0	5	10
5	16807	83	0	7	76
6	117649	860	0	11	829	20

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 DerivedLength. Using these functions, the above data can be generated as follows:

gap> SortArithmeticFunctionSizes(7,1,DerivedLength);
[ [ 0, 0 ], [ 1, 1 ] ]
gap> SortArithmeticFunctionSizes(7,2,DerivedLength);
[ [ 0, 0 ], [ 1, 2 ], [ 2, 0 ] ]
gap> SortArithmeticFunctionSizes(7,3,DerivedLength);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 2 ], [ 3, 0 ] ]
gap> SortArithmeticFunctionSizes(7,4,DerivedLength);
[ [ 0, 0 ], [ 1, 5 ], [ 2, 10 ], [ 3, 0 ], [ 4, 0 ] ]
gap> SortArithmeticFunctionSizes(7,5,DerivedLength);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 76 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(7,6,DerivedLength);
[ [ 0, 0 ], [ 1, 11 ], [ 2, 829 ], [ 3, 20 ], [ 4, 0 ], [ 5, 0 ], [ 6, 0 ] ]

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

$n$	$7^{n}$	total number of groups	length 0	length 1	length 2	length 3
0	1	1	1
1	7	1	0	1
2	49	2	0	1
3	343	5	0	0.6000	0.4000
4	2401	15	0	0.3333	0.6667
5	16807	83	0	0.0843	0.9157
6	117649	860	0	0.0128	0.9640	0.0233