# Groups of order 27

See pages on algebraic structures of order 27| See pages on groups of a particular order

## Statistics at a glance

Since $27 = 3^3$ is a prime power, and prime power order implies nilpotent, all the groups of order 27 are nilpotent groups.

Quantity Value Explanation
Total number of groups up to isomorphism 5
Number of abelian groups 3 equal to the number of unordered integer partitions of 3. See classification of finite abelian groups
Number of groups of nilpotency class exactly two 2

## The list

To learn more about how to come up with the list and prove that it is exhaustive (i.e., that these are precisely the isomorphism classes of groups of order 27), see classification of groups of prime-cube order
To understand these in a broader context, see
groups of order 3^n|groups of prime-cube order
Common name for group Second part of GAP ID (GAP ID is (27,second part)) Nilpotency class Probability in cohomology tree probability distribution (as proper fraction) Probability in cohomology tree probability distribution (as numerical value)
cyclic group:Z27 1 1 4/9 0.4444
direct product of Z9 and Z3 2 1 26/81 0.3210
prime-cube order group:U(3,3) 3 2 2/81 0.0247
M27 (semidirect product of Z9 and Z3) 4 2 16/81 0.1975
elementary abelian group:E27 5 1 1/81 0.0123

## Arithmetic functions

### Functions taking values between 0 and 3

Group GAP ID (second part) prime-base logarithm of exponent nilpotency class derived length Frattini length minimum size of generating set subgroup rank rank as p-group normal rank characteristic rank prime-base logarithm of order of derived subgroup prime-base logarithm of order of inner automorphism group
cyclic group:Z27 1 3 1 1 3 1 1 1 1 1 0 0
direct product of Z9 and Z3 2 2 1 1 2 2 2 2 2 2 0 0
prime-cube order group:U(3,3) 3 1 2 2 2 2 2 2 2 1 1 2
M27 4 2 2 2 2 2 2 2 2 2 1 2
elementary abelian group:E27 5 1 1 1 1 3 3 3 3 3 0 0
mean (with equal weight on all groups) -- 1.8 1.4 1.4 2 2 2 2 2 1.8 0.4 0.8
mean (weighting by cohomology tree probability distribution) -- 2.4074 1.2222 1.2222 2.4321 1.5679 1.5679 1.5679 1.5679 1.5432 0.2222 0.4444

Here is the same information, with rows and columns interchanged:

### Numerical invariants

Group Conjugacy class sizes degrees of irreducible representations
cyclic group:Z27 1 (27 times) 1 (27 times)
direct product of Z9 and Z3 1 (27 times) 1 (27 times)
prime-cube order group:U(3,3) 1 (3 times), 3 (8 times) 1 (9 times), 3 (2 times)
semidirect product of Z9 and Z3 1 (3 times), 3 (8 times) 1 (9 times), 3 (2 times)
elementary abelian group:E27 1 (27 times) 1 (27 times)

## Group properties

Property cyclic group:Z27 direct product of Z9 and Z3 prime-cube order group:U(3,3) semidirect product of Z9 and Z3 elementary abelian group:E27
cyclic group Yes No No No No
homocyclic group Yes No No No Yes
metacyclic group Yes Yes Yes Yes No
abelian group Yes Yes No No Yes
ambivalent group No No No No No
group of nilpotency class two Yes Yes Yes Yes Yes
group in which every element is automorphic to its inverse Yes Yes Yes No Yes
group in which any two elements generating the same cyclic subgroup are automorphic Yes Yes Yes No Yes