is the cyclic group that is the unique
group of group order 7.
Examples include the point group 
and the integers modulo 7 under addition (
). No modulo multiplication
group is isomorphic to 
. Like all cyclic groups, 
is Abelian.
The cycle graph is shown above, and the group has cycle index is
 |
The elements 
of the group satisfy 
,
where 1 is the identity element.
Its multiplication table is illustrated above and enumerated below.
 | 1 |  |  |  |  |  |  |
| 1 | 1 |  |  |  |  |  |  |
 |  |  |  |  |  |  | 1 |
 |  |  |  |  |  | 1 |  |
 |  |  |  |  | 1 |  |  |
 |  |  |  | 1 |  |  |  |
 |  |  | 1 |  |  |  |  |
 |  | 1 |  |  |  |  |  |
Because it is Abelian, the group conjugacy classes are 
,
, 
, 
, 
, 
, and 
. Because 7 is prime, the only subgroups are the trivial
group and the entire group. 
is therefore a simple group,
as are all cyclic graphs of prime order.
