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).

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.