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.