Groups of order 60

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:

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

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	The 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.

The list

Group	GAP ID (second part)	Abelian?
direct product of Z5 and Dic12	1	No
direct product of Z3 and Dic20	2	No
dicyclic group:Dic60	3	No
cyclic group:Z60	4	Yes
alternating group:A5	5	No
direct product of Z3 and GA(1,5)	6	No
semidirect product of Z3 and GA(1,5)	7	No
direct product of S3 and D10	8	No
direct product of Z5 and A4	9	No
direct product of Z6 and D10	10	No
direct product of Z10 and S3	11	No
dihedral group:D60	12	No
direct product of Z2 and Z30	13	Yes

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.