Groups of order 243

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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]