Groups of order 27

This article gives information about, and links to more details on, 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