Groups of order 7^n

Number of groups of small orders


Nilpotency class


Here is the GAP code to generate this information:

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.



Derived length


Here is the GAP code to generate this information:

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.

