Groups of order 243

This article gives information about, and links to more details on, groups of order 243
See pages on algebraic structures of order 243 | See pages on groups of a particular order

Statistics at a glance

Since $243=3^{5}$ is a prime power, and prime power order implies nilpotent, all groups of order 243 are nilpotent groups.

Quantity	Value	Explanation
Total number of groups up to isomorphism	67
Number of abelian groups	7	Equals the number of unordered integer partitions of 5, see classification of finite abelian groups
Number of groups of nilpotency class exactly two	28
Number of groups of nilpotency class exactly three	26
Number of groups of nilpotency class exactly four (i.e., maximal class groups)	6

Arithmetic functions

Summary information

Here, the rows are arithmetic functions that take values between $0$ and $5$, and the columns give the possible values of these functions. The entry in each cell is the number of isomorphism classes of groups for which the row arithmetic function takes the column value. Note that all the row value sums must equal $67$, which is the total number of groups of order $243$

Arithmetic function	Value 0	Value 1	Value 2	Value 3	Value 4	Value 5
prime-base logarithm of exponent	0	4	49	11	2	1
Frattini length	0	1	46	17	2	1
nilpotency class	0	7	28	26	6	0
derived length	0	7	60	0	0	0
minimum size of generating set	0	1	29	30	6	1
rank as p-group	0	1	15	42	8	1
normal rank as p-group	0	1	15	42	8	1
characteristic rank as p-group	0	1	17	39	7	1

Here is the GAP code to generate this information: [SHOW MORE]

We use the function SortArithmeticFunctionSizes and some in-built and easy-to-code functions. The in-built functions used are Exponent and Logarithm (for [prime-base logarithm of exponent]]), NilpotencyClassOfGroup (for nilpotency class), DerivedLength (for derived length), and Rank (for minimum size of generating set). The functions FrattiniLength (for Frattini length), RankAsPGroup (for rank of a p-group), NormalRank (for normal rank of a p-group)), and CharacteristicRank (for characteristic rank of a p-group) are not in-built but are easily coded (follow links to get code).

gap> SortArithmeticFunctionSizes(3,5,G -> Log(Exponent(G),3));
[ [ 0, 0 ], [ 1, 4 ], [ 2, 49 ], [ 3, 11 ], [ 4, 2 ], [ 5, 1 ] ]
gap> SortArithmeticFunctionSizes(3,5,FrattiniLength);
[ [ 0, 0 ], [ 1, 1 ], [ 2, 46 ], [ 3, 17 ], [ 4, 2 ], [ 5, 1 ] ]
gap> SortArithmeticFunctionSizes(3,5,NilpotencyClassOfGroup);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 28 ], [ 3, 26 ], [ 4, 6 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(3,5,DerivedLength);
[ [ 0, 0 ], [ 1, 7 ], [ 2, 60 ], [ 3, 0 ], [ 4, 0 ], [ 5, 0 ] ]
gap> SortArithmeticFunctionSizes(3,5,Rank);
[ [ 0, 0 ], [ 1, 1 ], [ 2, 29 ], [ 3, 30 ], [ 4, 6 ], [ 5, 1 ] ]
gap> SortArithmeticFunctionSizes(3,5,RankAsPGroup);
[ [ 0, 0 ], [ 1, 1 ], [ 2, 15 ], [ 3, 42 ], [ 4, 8 ], [ 5, 1 ] ]
gap> SortArithmeticFunctionSizes(3,5,NormalRank);
[ [ 0, 0 ], [ 1, 1 ], [ 2, 15 ], [ 3, 42 ], [ 4, 8 ], [ 5, 1 ] ]
gap> SortArithmeticFunctionSizes(3,5,CharacteristicRank);
[ [ 0, 0 ], [ 1, 3 ], [ 2, 17 ], [ 3, 39 ], [ 4, 7 ], [ 5, 1 ] ]