Groups of order 36

This article gives information about, and links to more details on, groups of order 36
See pages on algebraic structures of order 36 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 36. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 36
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 36
subgroup structure	subgroup structure of groups of order 36
linear representation theory	linear representation theory of groups of order 36 projective representation theory of groups of order 36 modular representation theory of groups of order 36
endomorphism structure, automorphism structure	endomorphism structure of groups of order 36
group cohomology	group cohomology of groups of order 36

Statistics at a glance

The number 36 has prime factorization $36=2^{2}\cdot 3^{2}$. There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$-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	List/comment
Number of groups up to isomorphism	14
Number of abelian groups	4	((number of abelian groups of order 4) = 2) times ((number of abelian groups of order 9) = 2). See classification of finite abelian groups and structure theorem for finitely generated abelian groups
Number of nilpotent groups	4	((number of groups of order 4) = 2) times ((number of groups of order 9) = 2). See equivalence of definitions of finite nilpotent group and classification of groups of prime-square order
Number of solvable groups	14	There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$-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

The list

There are 14 groups of order 36.

Group	Second part of GAP ID	Abelian	Direct Product
Dicyclic group:Dic36	1	no	no
Cyclic group:Z36	2	yes	yes
SmallGroup(36,3)	3	no	no
Dihedral group:D36	4	no	yes
Direct product of E4 and Z9	5	yes	yes
SmallGroup(36,6)	6	no	yes
SmallGroup(36,7)	7	no	no
Direct product of E9 and Z4	8	yes	yes
SmallGroup(36,9)	9	no	no
Direct product of S3 and S3	10	no	yes
Direct product of A4 and Z3	11	no	yes
Direct product of D12 and Z3	12	no	yes
SmallGroup(36,13)	13	no	yes
Direct product of E4 and E9	14	yes	yes

Endomorphism structure

Further information: Endomorphism structure of groups of order 36

Automorphism groups

The automorphism groups of the groups are as follows:

Group	Second part of GAP ID	Isomorphism class of automorphism group	Automorphism group order
Dicyclic group:Dic36	1	Direct product of Z2 and SmallGroup(54,6)	108
Cyclic group:Z36	2	Direct product of Z6 and Z2	12
SmallGroup(36,3)	3	Direct product of S4 and Z3	72
Dihedral group:D36	4	Direct product of Z2 and SmallGroup(54,6)	108
Direct product of E4 and Z9	5	Direct product of D12 and Z3	36
SmallGroup(36,6)	6	Direct product of D12 and Z2	24
SmallGroup(36,7)	7	Direct product of Z2 and GA(2,3)	864
Direct product of E9 and Z4	8	Direct product of Z2 and GL(2,3)	96
SmallGroup(36,9)	9	General semiaffine group:GammaA(1,9)	144
Direct product of S3 and S3	10	Wreath product of S3 and Z2	72
Direct product of A4 and Z3	11	Direct product of S3 and S4	144
SmallGroup(36,12)	12	Direct product of D12 and Z2	24
SmallGroup(36,13)	13	Direct product of Z2 and GA(2,3)	864
Direct product of E4 and E9	14	Direct product of S3 and GL(2,3)	288

GAP implementation

The order 36 is part of GAP's SmallGroup library. Hence, any group of order 36 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 36 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(36);

  There are 14 groups of order 36.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 6, 1 ].
     2 has Frattini factor [ 6, 2 ].
     3 has Frattini factor [ 12, 3 ].
     4 has Frattini factor [ 12, 4 ].
     5 has Frattini factor [ 12, 5 ].
     6 has Frattini factor [ 18, 3 ].
     7 has Frattini factor [ 18, 4 ].
     8 has Frattini factor [ 18, 5 ].
     9 - 14 have trivial Frattini subgroup.

  For the selection functions the values of the following attributes
  are precomputed and stored:
     IsAbelian, IsNilpotentGroup, IsSupersolvableGroup, IsSolvableGroup,
     LGLength, FrattinifactorSize and FrattinifactorId.

  This size belongs to layer 2 of the SmallGroups library.
  IdSmallGroup is available for this size.