Groups of order 12

This article gives information about, and links to more details on, groups of order 12
See pages on algebraic structures of order 12| See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 12. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 12
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 12
subgroup structure	subgroup structure of groups of order 12
linear representation theory	linear representation theory of groups of order 12 projective representation theory of groups of order 12 modular representation theory of groups of order 12
endomorphism structure, automorphism structure	endomorphism structure of groups of order 12
group cohomology	group cohomology of groups of order 12

Statistics at a glance

The number 12 has prime factorization $12 = 2^2 \cdot 3$. There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity	Value	Explanation
Total number of groups	5
Number of abelian groups	2	(number of abelian groups of order $2^2$) times (number of abelian groups of order $3^1$) = (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) = $2 \times 1 = 2$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups	2	(number of groups of order 4) times (number of groups of order 3) = $2 \times 1 = 2$ See number of nilpotent groups equals product of number of groups of order each maximal prime power divisor, which in turn follows from equivalence of definitions of finite nilpotent group.
Number of solvable groups	5	There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^aq^b$-theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
Number of simple groups	0	Follows from all groups of this order being solvable.

CLASSIFICATION: For an explanation of how to arrive at this list of groups, and prove that it is comprehensive, see classification of groups of order 12.

The list

Group	Second part of GAP ID (GAP ID is (12,second part))	Abelian?	2-Sylow subgroup	Is the 2-Sylow subgroup normal?	Is the 3-Sylow subgroup normal?
dicyclic group:Dic12	1	No	cyclic group:Z4	No	Yes
cyclic group:Z12	2	Yes	cyclic group:Z4	Yes	Yes
alternating group:A4	3	No	Klein four-group	Yes	No
dihedral group:D12	4	No	Klein four-group	No	Yes
direct product of Z6 and Z2	5	Yes	Klein four-group	Yes	Yes

Arithmetic functions

Function	dicyclic group:Dic12	cyclic group:Z12	alternating group:A4	dihedral group:D12	direct product of Z6 and Z2
exponent	12	12	6	6	6
derived length	2	1	2	2	1
Fitting length	2	1	2	2	1
nilpotency class	--	1	--	--	1
minimum size of generating set	2	1	2	2	2
subgroup rank	2	1	2	2	2
number of conjugacy classes	6	12	4	6	12

Numerical invariants

Group	Conjugacy class sizes	degrees of irreducible representations
dicyclic group:Dic12	1,1,2,2,3,3	1,1,1,1,2,2
cyclic group:Z12	1 (12 times)	1 (12 times)
alternating group:A4	1,3,4,4	1,1,1,3
dihedral group:D12	1,1,2,2,3,3	1,1,1,1,2,2
direct product of Z6 and Z2	1 (12 times)	1 (12 times)

Group properties

Function	dicyclic group:Dic12	cyclic group:Z12	alternating group:A4	dihedral group:D12	direct product of Z6 and Z2
cyclic group	No	Yes	No	No	No
abelian group	No	Yes	No	No	Yes
metacyclic group	Yes	Yes	No	Yes	Yes
nilpotent group	No	Yes	No	No	Yes
supersolvable group	Yes	Yes	No	Yes	Yes
solvable group	Yes	Yes	Yes	Yes	Yes
T-group	Yes	Yes	No	Yes	Yes

Sylow subgroups

2-Sylow subgroups

Here is the occurrence summary:

Group of order 4	GAP ID (second part)	Number of groups of order 12 in which it is a 2-Sylow subgroup	List of these groups	Second part of GAP IDs of these groups
cyclic group:Z4	1	2	dicyclic group:Dic12, cyclic group:Z12	1,2
Klein four-group	2	3	alternating group:A4, dihedral group:D12, direct product of Z6 and Z2	3,4,5

Note that the number of 2-Sylow subgroups is either 1 or 3. The former happens if and only if we have a normal Sylow subgroup for the prime 2. The latter happens if and only if we have a self-normalizing Sylow subgroup for the prime 2.

Group	Second part of GAP ID (ID is (12,second part))	2-Sylow subgroup	Second part of GAP ID	Number of 2-Sylow subgroups	Subgroup structure page
dicyclic group:Dic12	1	cyclic group:Z4	1	3	subgroup structure of dicyclic group:Dic12, subgroup structure of dicyclic groups
cyclic group:Z12	2	cyclic group:Z4	1	1	subgroup structure of cyclic group:Z12, subgroup structure of cyclic groups
alternating group:A4	3	Klein four-group	2	1	subgroup structure of alternating group:A4, subgroup structure of alternating groups
dihedral group:D12	4	Klein four-group	2	3	subgroup structure of dihedral group:D12, subgroup structure of dihedral groups
direct product of Z6 and Z2	5	Klein four-group	2	1	subgroup structure of direct product of Z6 and Z2, subgroup structure of finite abelian groups

The only non-abelian (and also the only non-nilpotent) example where there is a normal Sylow subgroup is the case of alternating group:A4.

GAP implementation

The order 12 is part of GAP's SmallGroup library. Hence, any group of order 12 can be constructed using the SmallGroup function by specifying its group ID. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Further, the collection of all groups of order 12 can be accessed as a list using GAP's AllSmallGroups function.

Here is GAP's summary information about how it stores groups of this order, accessed using GAP's SmallGroupsInformation function:

gap> SmallGroupsInformation(12);

  There are 5 groups of order 12.
    1 is of type 6.2.
    2 is of type c12.
    3 is of type A4.
    4 is of type D12.
    5 is of type 2^2x3.

  The groups whose order factorises in at most 3 primes
  have been classified by O. Hoelder. This classification is
  used in the SmallGroups library.

  This size belongs to layer 1 of the SmallGroups library.
  IdSmallGroup is available for this size.