# Groups of order 36

## Contents

See pages on algebraic structures of order 36| See pages on groups of a particular order

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 $p^aq^b$-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 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 $p^aq^b$-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

## The list

There are 15 groups of order 36.

Group Second part of GAP ID Abelian Direct Product
SmallGroup(36,1) 1 no no
Cyclic group:Z36 2 yes yes
SmallGroup(36,3) 3 no no
SmallGroup(36,4) 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
SmallGroup(36,12) 12 no yes
SmallGroup(36,13) 13 no yes
Direct product of E4 and E9 14 yes yes

## 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.