# Groups of order 60

## Contents

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

Other expressions for this number are: $60 = 5!/2 = 4^3 - 4 = (5^3 - 5)/2$

### Group counts

Quantity Value Explanation
Total number of groups up to isomorphism 13
Number of abelian groups (i.e., finite abelian groups) up to isomorphism 2 (number of abelian groups of order $2^2$) times (number of abelian groups of order $3^1$) times (number of abelian groups of order $5^1$) = (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) times (number of unordered integer partitions of 1) = $2 \times 1 \times 1 = 2$. 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 2 (number of groups of order 4) times (number of groups of order 3) times (number of groups of order 5) = $2 \times 1 \times 1 = 2$. 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 12 See note on non-solvable groups
Number of non-solvable groups up to isomorphism 1 Th only possibility is alternating group:A5; see below
Number of simple groups up to isomorphism 1 The unique such group is alternating group:A5. See A5 is simple and A5 is the simple non-abelian group of smallest order.
Number of almost simple groups up to isomorphism 1 alternating group:A5
Number of quasisimple groups up to isomorphism 1 alternating group:A5
Number of almost quasisimple groups up to isomorphism 1 alternating group:A5
Number of perfect groups up to isomorphism 1 alternating group:A5

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.