# Groups of order 660

## Contents

See pages on algebraic structures of order 660| See pages on groups of a particular order

## Statistics at a glance

The number 660 has the prime factorization: $\! 660 = 2^2 \cdot 3^1 \cdot 5^1 \cdot 11^1 = 4 \cdot 3 \cdot 5 \cdot 11$

There are both solvable and non-solvable groups of this order (see the table below).

Quantity Value List/comment
Total number of groups 41
Number of abelian groups, i.e., finite abelian groups 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 $5^1$) times (number of abelian groups of order $11^1$) = (number of unordered integer partitions of 2) times (number of unordered integer partitions of 1) times (number of unordered integer partitions of 1) times (number of unordered integer partitions of 1) = $2 \times 1 \times 1 \times 1 = 2$. See classification of finite abelian groups and structure theorem for finitely generated abelian groups.
Number of nilpotent groups, i.e., finite nilpotent groups 2 ((number of groups of order 4) times (number of groups of order 3) times (number of groups of order 5) times (number of groups of order 11) = $2 \times 1 \times 1 \times 1 = 2$. 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 supersolvable groups, i.e., finite supersolvable groups 36
Number of solvable groups 38 See note on non-solvable groups
Number of non-solvable groups 2 The only non-solvable groups are PSL(2,11) (ID: (660,13)) and direct product of A5 and Z11 (ID: (660,14)).
Number of simple groups 1 PSL(2,11) is the only simple group.

## GAP implementation

The order 660 is part of GAP's SmallGroup library. Hence, any group of order 660 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 660 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(660);

There are 40 groups of order 660.
They are sorted by their Frattini factors.
1 has Frattini factor [ 330, 1 ].
2 has Frattini factor [ 330, 2 ].
3 has Frattini factor [ 330, 3 ].
4 has Frattini factor [ 330, 4 ].
5 has Frattini factor [ 330, 5 ].
6 has Frattini factor [ 330, 6 ].
7 has Frattini factor [ 330, 7 ].
8 has Frattini factor [ 330, 8 ].
9 has Frattini factor [ 330, 9 ].
10 has Frattini factor [ 330, 10 ].
11 has Frattini factor [ 330, 11 ].
12 has Frattini factor [ 330, 12 ].
13 - 40 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.