View source for Groups of order 27

{{groups of order|27}}

==Statistics at a glance==

Since <math>27 = 3^3</math> is a [[prime power]], and [[prime power order implies nilpotent]], all the groups of order 27 are nilpotent groups.

{| class="sortable" border="1"
! 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==

{{quotation|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]]}}

{{quotation|To understand these in a broader context, see<br>[[groups of order 3^n]]<nowiki>|</nowiki>[[groups of prime-cube order]]}}

{| class="sortable" border="1"
! 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===

{| class="sortable" border="1"
! Group !! GAP ID (second part) !! [[prime-base logarithm of exponent]] !![[nilpotency class]] !! [[derived length]] !! [[Frattini length]] !! [[minimum size of generating set]] !! [[subgroup rank of a group|subgroup rank]] !! [[rank of a p-group|rank as p-group]] !! [[normal rank of a p-group|normal rank]] !! [[characteristic rank of a p-group|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:

===Arithmetic functions values of a counting nature===

===Numerical invariants===

{| class="wikitable" border="1"
! 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==

{| class="wikitable" border="1"
! 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
|}