Groups of order 72

This article gives information about, and links to more details on, groups of order 72
See pages on algebraic structures of order 72 | See pages on groups of a particular order

This article gives basic information comparing and contrasting groups of order 72. The prime factorization is $72=2^{3}\cdot 3^{2}$. Since the order has only two prime factors, and order has only two prime factors implies solvable, all groups of this order are solvable groups (and in particular, finite solvable groups).

Statistics at a glance

Quantity	Value	Explanation
Number of groups of order 72	50
Number of abelian groups	6	(number of groups of order $2^{3}$) times (number of groups of order $3^{2}$) = (number of unordered integer partitions of 3) times (number of unordered integer partitions of 2) = $3\times 2=6$. See classification of finite abelian groups
Number of nilpotent groups	10	(number of groups of order 8) times (number of groups of order 9) = $5\times 2=10$. 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	50	Since the order has only two prime factors, and order has only two prime factors implies solvable, all groups of this order are solvable groups (and in particular, finite solvable groups).
Number of simple groups	0	Follows from the fact that all groups of the order are solvable.

GAP implementation

The order 72 is part of GAP's SmallGroup library. Hence, any group of order 72 can be constructed using the SmallGroup function by specifying its group ID. Unfortunately, IdGroup is not available for this order, i.e., given a group of this order, it is not possible to directly query GAP to find its GAP ID.

Further, the collection of all groups of order 72 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(72);

  There are 50 groups of order 72.
  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 has Frattini factor [ 18, 3 ].
     13 has Frattini factor [ 18, 4 ].
     14 has Frattini factor [ 18, 5 ].
     15 has Frattini factor [ 24, 12 ].
     16 has Frattini factor [ 24, 13 ].
     17 has Frattini factor [ 24, 14 ].
     18 has Frattini factor [ 24, 15 ].
     19 has Frattini factor [ 36, 9 ].
     20 - 24 have Frattini factor [ 36, 10 ].
     25 has Frattini factor [ 36, 11 ].
     26 - 30 have Frattini factor [ 36, 12 ].
     31 - 35 have Frattini factor [ 36, 13 ].
     36 - 38 have Frattini factor [ 36, 14 ].
     39 - 50 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.