Groups of order 216

This article gives information about, and links to more details on, groups of order 216
See pages on algebraic structures of order 216 | See pages on groups of a particular order

Statistics at a glance

The number 216 has prime factorization $216=2^{3}\cdot 3^{3}$ . There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$ -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.

Quantity	Value	Explanation
Total number of groups	177
Number of abelian groups	9	(number of abelian groups of order $2^{3}$ ) times (number of abelian groups of order $3^{3}$ ) = (number of unordered integer partitions of 3) times (number of unordered integer partitions of 3) = $3\times 3=9$ . See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups	25	(number of groups of order 8) times (number of groups of order 27) = $5\times 5=25$ . 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	177	There are only two prime factors of this number. Order has only two prime factors implies solvable (by Burnside's $p^{a}q^{b}$ -theorem) and hence all groups of this order are solvable groups (specifically, finite solvable groups). Another way of putting this is that the order is a solvability-forcing number. In particular, there is no simple non-abelian group of this order.
Number of simple groups	0

GAP implementation

The order 216 is part of GAP's SmallGroup library. Hence, any group of order 216 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 216 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(216);

  There are 177 groups of order 216.
  They are sorted by their Frattini factors.
     1 has Frattini factor [ 6, 1 ].
     2 has Frattini factor [ 6, 2 ].
     3 has Frattini factor [ 12, 3 ].
     4 - 8 have Frattini factor [ 12, 4 ].
     9 - 11 have Frattini factor [ 12, 5 ].
     12 - 15 have Frattini factor [ 18, 3 ].
     16 - 17 have Frattini factor [ 18, 4 ].
     18 - 20 have Frattini factor [ 18, 5 ].
     21 has Frattini factor [ 24, 12 ].
     22 has Frattini factor [ 24, 13 ].
     23 has Frattini factor [ 24, 14 ].
     24 has Frattini factor [ 24, 15 ].
     25 has Frattini factor [ 36, 9 ].
     26 - 37 have Frattini factor [ 36, 10 ].
     38 - 42 have Frattini factor [ 36, 11 ].
     43 - 62 have Frattini factor [ 36, 12 ].
     63 - 72 have Frattini factor [ 36, 13 ].
     73 - 81 have Frattini factor [ 36, 14 ].
     82 has Frattini factor [ 54, 12 ].
     83 has Frattini factor [ 54, 13 ].
     84 has Frattini factor [ 54, 14 ].
     85 has Frattini factor [ 54, 15 ].
     86 has Frattini factor [ 72, 39 ].
     87 has Frattini factor [ 72, 40 ].
     88 has Frattini factor [ 72, 41 ].
     89 - 92 have Frattini factor [ 72, 42 ].
     93 - 95 have Frattini factor [ 72, 43 ].
     96 - 99 have Frattini factor [ 72, 44 ].
     100 has Frattini factor [ 72, 45 ].
     101 - 102 have Frattini factor [ 72, 46 ].
     103 - 107 have Frattini factor [ 72, 47 ].
     108 - 111 have Frattini factor [ 72, 48 ].
     112 - 113 have Frattini factor [ 72, 49 ].
     114 - 116 have Frattini factor [ 72, 50 ].
     117 has Frattini factor [ 108, 36 ].
     118 has Frattini factor [ 108, 37 ].
     119 - 123 have Frattini factor [ 108, 38 ].
     124 - 130 have Frattini factor [ 108, 39 ].
     131 - 133 have Frattini factor [ 108, 40 ].
     134 has Frattini factor [ 108, 41 ].
     135 - 139 have Frattini factor [ 108, 42 ].
     140 - 144 have Frattini factor [ 108, 43 ].
     145 - 149 have Frattini factor [ 108, 44 ].
     150 - 152 have Frattini factor [ 108, 45 ].
     153 - 177 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.