# Groups of order 60

From Groupprops

This article gives information about, and links to more details on, groups of order 60

See pages on algebraic structures of order 60| See pages on groups of a particular order

## Statistics at a glance

### Factorization and useful forms

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

Other expressions for this number are:

### Group counts

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

## GAP implementation

The order 60 is part of GAP's SmallGroup library. Hence, any group of order 60 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 60 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(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.