The cyclic group is one of the two Abelian groups of the five groups total of group order 12 (the other order-12 Abelian group being finite group C2×C6). Examples include the modulo multiplication groups of orders and 26 (which are the only modulo multiplication groups isomorphic to ).

The cycle graph of is shown above. The cycle index is

Its multiplication table is illustrated above.

The numbers of elements satisfying for , 2, ..., 12 are 1, 2, 3, 4, 1, 6, 1, 4, 3, 2, 1, 12.

Because the group is Abelian, each element is in its own conjugacy class. There are six subgroups: , , , and . , 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.