# Groups of order 81

## Contents

See pages on algebraic structures of order 81| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order $81$. See also more detailed information on specific subtopics through the links:

Information type Page summarizing information for groups of order 81
element structure element structure of groups of order 81
subgroup structure subgroup structure of groups of order 81
linear representation theory linear representation theory of groups of order 81

## Statistics at a glance

Quantity Value
Total number of groups 15
Number of abelian groups 5
Number of groups of class exactly two 6
Number of groups of class exactly three 4

## The list

To understand these in a broader context, see
groups of order 3^n|groups of prime-fourth order
Common name for group Second part of GAP ID (GAP ID is (p^4, second part)) Nilpotency class
Cyclic group:Z81 1 1
Direct product of Z9 and Z9 2 1
SmallGroup(81,3) 3 2
Nontrivial semidirect product of Z9 and Z9 4 2
Direct product of Z27 and Z3 5 1
M81 6 2
Wreath product of Z3 and Z3 7 3
SmallGroup(81,8) 8 3
SmallGroup(81,9) 9 3
SmallGroup(81,10) 10 3
Direct product of Z9 and E9 11 1
Direct product of prime-cube order group:U(3,3) and Z3 12 2
Direct product of semidirect product of Z9 and Z3 and Z3 13 2
Central product of prime-cube order group:U(3,3) and Z9 14 2
Elementary abelian group:E81 15 1

## Arithmetic functions

### Summary information

Here, rows are arithmetic functions taking values between 0 and 4, 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 this column value. Note that all the row sums must equal 15, which is the total number of groups of order 81.

Arithmetic function Value 0 Value 1 Value 2 Value 3 Value 4
prime-base logarithm of exponent 0 2 8 4 1
nilpotency class 0 5 6 4 0
derived length 0 5 10 0 0
Frattini length 0 1 11 2 1
minimum size of generating set 0 1 9 4 1
subgroup rank 0 1 7 6 1
rank 0 1 8 5 1
normal rank 0 1 8 5 1
characteristic rank 0 2 7 4 1
prime-base logarithm of order of derived subgroup 5 6 4 0 0
prime-base logarithm of order of inner automorphism group 5 0 6 4 0

### Functions taking values between 0 and 4

These measure ranks of subgroups, lengths of series, or the prime-base logarithms of orders of certain subgroups.

Group GAP ID second part prime-base logarithm of exponent nilpotency class derived length Frattini length minimum size of generating set subgroup rank rank normal rank characteristic rank prime-base logarithm of order of derived subgroup prime-base logarithm of order of inner automorphism group
Cyclic group:Z81 1 4 1 1 4 1 1 1 1 1 0 0
Direct product of Z9 and Z9 2 2 1 1 2 2 2 2 2 2 0 0
SmallGroup(81,3) 3 2 2 2 2 2 3 3 3 3 1 2
Nontrivial semidirect product of Z9 and Z9 4 2 2 2 2 2 2 2 2 2 1 2
Direct product of Z27 and Z3 5 3 1 1 3 2 2 2 2 2 0 0
Semidirect product of Z27 and Z3 6 3 2 2 3 2 2 2 2 2 1 2
Wreath product of Z3 and Z3 7 2 3 2 2 2 3 3 3 3 2 3
SmallGroup(81,8) 8 2 3 2 2 2 2 2 2 2 2 3
SmallGroup(81,9) 9 3 3 2 2 2 2 2 2 2 2 3
SmallGroup(81,10) 10 3 3 2 2 2 2 2 2 2 2 3
Direct product of Z9 and E9 11 2 1 1 2 3 3 3 3 3 0 0
Direct product of prime-cube order group:U(3,3) and Z3 12 1 2 2 2 3 3 3 3 2 1 2
Direct product of semidirect product of Z9 and Z3 and Z3 13 2 2 2 2 3 3 3 3 3 1 2
SmallGroup(81,14) 14 2 2 2 2 3 3 2 2 1 1 2
Elementary abelian group:E81 15 1 1 1 1 4 4 4 4 4 0 0