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

This article gives basic information comparing and contrasting groups of order 120. See also more detailed information on specific subtopics through the links:

Information type	Page summarizing information for groups of order 120
element structure (element orders, conjugacy classes, etc.)	element structure of groups of order 120
subgroup structure	subgroup structure of groups of order 120
linear representation theory	linear representation theory of groups of order 120 projective representation theory of groups of order 120 modular representation theory of groups of order 120
endomorphism structure, automorphism structure	endomorphism structure of groups of order 120
group cohomology	group cohomology of groups of order 120

Statistics at a glance

The number 120 has prime factorization $120 = 2^{3} \cdot 3 \cdot 5$ . There are solvable as well as non-solvable groups of this order. All the non-solvable groups have alternating group:A5 and cyclic group:Z2 as the composition factors in their composition series.

Quantity	Value	List/comment
Total number of groups	47
Number of abelian groups	3	(number of abelian groups of order $2^{3}$ ) times (number of abelian groups of order $3^{1}$ ) times (number of abelian groups of order $5^{1}$ ) = $3 \times 1 \times 1 = 3$ . See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups	5	(number of groups of order 8) times (number of groups of order 3) times (number of groups of order 5) = $5 \times 1 \times 1 = 5$ . 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	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 $P G L (2, 5)$ )
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.