# Groups of order 180

## Contents

See pages on algebraic structures of order 180| See pages on groups of a particular order

Information type Page summarizing information for groups of order 180
element structure (element orders, conjugacy classes, etc.) element structure of groups of order 180
subgroup structure subgroup structure of groups of order 180
linear representation theory linear representation theory of groups of order 180
projective representation theory of groups of order 180
modular representation theory of groups of order 180
endomorphism structure, automorphism structure endomorphism structure of groups of order 180
group cohomology group cohomology of groups of order 180

## Statistics at a glance

The number 180 has prime factorization $180 = 2^2 \cdot 3^2 \cdot 5$. There are both solvable and non-solvable groups of this order. The only possible non-abelian composition factor for this order is alternating group:A5.

Quantity Value Explanation
Total number of groups 37
Number of abelian groups 4 (number of abelian groups of order $2^2$) times (number of abelian groups of order $3^2$) times (number of abelian groups of order $5^1$) = (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) = $2 \times 2 \times 1 = 4$. 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) times (number of groups of order 9) times (number of groups of order 5) = $2 \times 2 \times 1 = 4$. 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 36 The only non-solvable group is direct product of A5 and Z3, also described as the general linear group of degree two over field:F4, i.e., $GL(2,4)$.
Number of simple groups 0

## GAP implementation

The order 180 is part of GAP's SmallGroup library. Hence, any group of order 180 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 180 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(180);

There are 37 groups of order 180.
They are sorted by their Frattini factors.
1 has Frattini factor [ 30, 1 ].
2 has Frattini factor [ 30, 2 ].
3 has Frattini factor [ 30, 3 ].
4 has Frattini factor [ 30, 4 ].
5 has Frattini factor [ 60, 6 ].
6 has Frattini factor [ 60, 7 ].
7 has Frattini factor [ 60, 8 ].
8 has Frattini factor [ 60, 9 ].
9 has Frattini factor [ 60, 10 ].
10 has Frattini factor [ 60, 11 ].
11 has Frattini factor [ 60, 12 ].
12 has Frattini factor [ 60, 13 ].
13 has Frattini factor [ 90, 5 ].
14 has Frattini factor [ 90, 6 ].
15 has Frattini factor [ 90, 7 ].
16 has Frattini factor [ 90, 8 ].
17 has Frattini factor [ 90, 9 ].
18 has Frattini factor [ 90, 10 ].
19 - 37 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.