Groups of order 360

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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

Factorization and useful forms

The number 360 has prime factorization:

360 = 2^3 \cdot 3^2 \cdot 5^1 = 8 \cdot 9 \cdot 5

Other expressions for this number are:

360 = 6!/2 = (9^3 - 9)/2 = 2^4(2^2 - 1)(2^4 - 1)/2

Group counts

Quantity Value Explanation
Number of groups up to isomorphism 162
Number of abelian groups (i.e., finite abelian groups) up to isomorphism 6 (Number of abelian groups of order 2^3) times (Number of abelian groups of order 3^2) times (Number of abelian groups of order 5^1) = (number of unordered integer partitions of 3) times (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) = 3 \times 2 \times 1 = 6. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups (i.e., finite nilpotent groups) up to isomorphism 10 (Number of groups of order 8) times (Number of groups of order 9) times (Number of groups of order 5) = 5 \times 2 \times 1 = 10. 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 (i.e., finite solvable groups) up to isomorphism 156 See note on non-solvable groups
Number of possibilities for multiset of composition factors, i.e., number of equivalence classes up to composition factor-equivalence 3 One equivalence class for finite solvable groups: cyclic group:Z2 (3 times), cyclic group:Z3 (2 times), cyclic group:Z5 (1 time)
Non-solvable equivalence class with just one piece: alternating group:A6 (order 360, 1 time)
Other non-solvable equivalence class: alternating group:A5 (order 60, 1 time), cyclic group:Z2 (1 time), cyclic group:Z3 (1 time).
See order of group is product of orders of composition factors and classification of possible multisets of composition factors for groups of a given order.
Number of non-solvable groups up to isomorphism 6 alternating group:A6 is one non-solvable group. All the others are groups that have alternating group:A5 (order 60) as the simple non-abelian composition factor and cyclic group:Z2 (1 time) and cyclic group:Z3 (1 time) as the other composition factors. There are five such groups.
Number of simple groups up to isomorphism 1 alternating group:A6 is the only simple group of this order
Number of quasisimple groups up to isomorphism 1 alternating group:A6
Number of almost simple groups up to isomorphism 1 alternating group:A6

GAP implementation

The order 360 is part of GAP's SmallGroup library. Hence, any group of order 360 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 360 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(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.