is one of the two groups of group order 6 which, unlike 
, is Abelian. It is also a cyclic. It is isomorphic to 
. Examples include the point groups 
and 
, the integers modulo 6 under addition (
), and the modulo multiplication groups 
, 
, and 
(with no others).
The cycle graph is shown above and has cycle index
 |
The elements 
of the group satisfy 
, where 1 is the identity element, three elements satisfy 
, and two elements satisfy 
.
Its multiplication table is illustrated above and enumerated below.
 | 1 |  |  |  |  |  |
| 1 | 1 |  |  |  |  |  |
 |  |  |  |  |  | 1 |
 |  |  |  |  | 1 |  |
 |  |  |  | 1 |  |  |
 |  |  | 1 |  |  |  |
 |  | 1 |  |  |  |  |
Since 
is Abelian, the conjugacy classes are 
, 
, 
, 
, 
, and 
. There are four subgroups of 
: 
, 
, 
, and 
which, because the group is Abelian, are all normal. Since 
has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.
