Groups of order 120

This article gives information about, and links to more details on, groups of order 120
See pages on algebraic structures of order 120 | See pages on groups of a particular order

Statistics at a glance

Quantity	Value	List/comment
Total number of groups	47
Number of abelian groups	3	((Number of abelian groups of order 8) = 3) times ((number of abelian groups of order 3) = 1) times ((number of abelian groups of order 5) = 1). See also classification of finite abelian groups and structure theorem for finitely generated abelian groups
Number of nilpotent groups	5	((Number of groups of order 8) = 5) times ((number of groups of order 3) = 1) times ((number of groups of order 5) = 1). See also equivalence of definitions of finite nilpotent group
Number of solvable groups	44	The three non-solvable groups are special linear group:SL(2,5), direct product of A5 and Z2, and symmetric group:S5
Number of simple groups	0
Number of almost simple groups	1	symmetric group:S5 (isomorphic to $PGL(2,5)$)
Number of quasisimple groups	1	special linear group:SL(2,5) (also the binary icosahedral group)

GAP implementation

The order 120 is part of GAP's SmallGroup library. Hence, any group of order 120 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 120 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(120);

  There are 47 groups of order 120.
  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, 5 ].
     6 has Frattini factor [ 60, 6 ].
     7 has Frattini factor [ 60, 7 ].
     8 - 14 have Frattini factor [ 60, 8 ].
     15 has Frattini factor [ 60, 9 ].
     16 - 20 have Frattini factor [ 60, 10 ].
     21 - 25 have Frattini factor [ 60, 11 ].
     26 - 30 have Frattini factor [ 60, 12 ].
     31 - 33 have Frattini factor [ 60, 13 ].
     34 - 47 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.