Groups of order 60

Factorization and useful forms
The number 60 has prime factors 2,3,5, and prime factorization:

$$60 = 2^2\cdot 3^1 \cdot 5^1 = 4\cdot 3\cdot 5$$

Other expressions for this number are:

$$60 = 5!/2 = 4^3 - 4 = (5^3 - 5)/2$$

Group counts
60 is the smallest possible order of a simple non-abelian group.

GAP implementation
gap> SmallGroupsInformation(60);

There are 13 groups of order 60. 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 - 13 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.