is the unique group of group order 3. It is both Abelian and cyclic. Examples include the point groups , , and and the integers under addition modulo 3 (). No modulo multiplication groups are isomorphic to .

The cycle graph of is shown above, and the cycle index is

The elements of the group satisfy where 1 is the identity element.

Its multiplication table is illustrated above and enumerated below (Cotton 1990, p. 10).

1 | |||

1 | 1 | ||

1 | |||

1 |

Since is Abelian, the conjugacy classes are , , and . The only subgroups of are the trivial group and the entire group, which are both trivially normal. is therefore a simple group, as are all cyclic graphs of prime order.

The irreducible representation (character table) is therefore

1 | |||

1 | 1 | 1 | |

1 | 1 | ||

1 | 1 |