Groups of order 180

This article gives information about, and links to more details on, groups of order 180
See pages on algebraic structures of order 180 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 180. See also more detailed information on specific subtopics through the links:

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.