Groups of order 360

This article gives information about, and links to more details on, groups of order 360
See pages on algebraic structures of order 360 | See pages on groups of a particular order

Statistics at a glance

The number 360 has prime factorization $360=2^{3}\cdot 3^{2}\cdot 5$. There are both solvable and non-solvable groups of this order.

Quantity	Value	Explanation
Number of groups up to isomorphism	162
Number of abelian groups up to isomorphism	6
Number of nilpotent groups up to isomorphism	10
Number of solvable groups up to isomorphism	156
Number of simple groups up to isomorphism	1
Number of quasisimple groups up to isomorphism	1
Number of almost simple groups up to isomorphism	1

GAP implementation

The order 360 is part of GAP's SmallGroup library. Hence, all groups of order 360 can be constructed using the SmallGroup function and have group IDs. Also, IdGroup is available, so the group ID of any group of this order can be queried.

Here is GAP's summary information about how it stores groups of this order:

gap> SmallGroupsInformation(360);

  There are 162 groups of order 360.
  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 - 13 have Frattini factor [ 60, 8 ].
     14 has Frattini factor [ 60, 9 ].
     15 - 19 have Frattini factor [ 60, 10 ].
     20 - 24 have Frattini factor [ 60, 11 ].
     25 - 29 have Frattini factor [ 60, 12 ].
     30 - 32 have Frattini factor [ 60, 13 ].
     33 has Frattini factor [ 90, 5 ].
     34 has Frattini factor [ 90, 6 ].
     35 has Frattini factor [ 90, 7 ].
     36 has Frattini factor [ 90, 8 ].
     37 has Frattini factor [ 90, 9 ].
     38 has Frattini factor [ 90, 10 ].
     39 has Frattini factor [ 120, 36 ].
     40 has Frattini factor [ 120, 37 ].
     41 has Frattini factor [ 120, 38 ].
     42 has Frattini factor [ 120, 39 ].
     43 has Frattini factor [ 120, 40 ].
     44 has Frattini factor [ 120, 41 ].
     45 has Frattini factor [ 120, 42 ].
     46 has Frattini factor [ 120, 43 ].
     47 has Frattini factor [ 120, 44 ].
     48 has Frattini factor [ 120, 45 ].
     49 has Frattini factor [ 120, 46 ].
     50 has Frattini factor [ 120, 47 ].
     51 has Frattini factor [ 180, 19 ].
     52 has Frattini factor [ 180, 20 ].
     53 has Frattini factor [ 180, 21 ].
     54 has Frattini factor [ 180, 22 ].
     55 has Frattini factor [ 180, 23 ].
     56 has Frattini factor [ 180, 24 ].
     57 has Frattini factor [ 180, 25 ].
     58 - 64 have Frattini factor [ 180, 26 ].
     65 - 71 have Frattini factor [ 180, 27 ].
     72 - 76 have Frattini factor [ 180, 28 ].
     77 - 83 have Frattini factor [ 180, 29 ].
     84 - 88 have Frattini factor [ 180, 30 ].
     89 has Frattini factor [ 180, 31 ].
     90 - 94 have Frattini factor [ 180, 32 ].
     95 - 99 have Frattini factor [ 180, 33 ].
     100 - 104 have Frattini factor [ 180, 34 ].
     105 - 109 have Frattini factor [ 180, 35 ].
     110 - 114 have Frattini factor [ 180, 36 ].
     115 - 117 have Frattini factor [ 180, 37 ].
     118 - 162 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.