Finite Group C_2×C_6

The finite group C_2×C_6 is the finite group of order 12 that is the group direct product of the cyclic group C2 and cyclic group C6. It is one of the two Abelian groups of order 12, the other being the cyclic group C12. Examples include the modulo multiplication groups M_(21), M_(28), M_(36), and M_(42) (and no other modulo multiplication groups).


The multiplication table is illustrated above. The numbers of elements for which A_i^n=1 for n=1, 2, ..., 12 are 1, 4, 3, 4, 1, 12, 1, 4, 3, 4, 1, 12.

Each element of C_2×C_6 is in its own conjugacy class. There are 10 subgroups: the trivial subgroup, 3 of length 2, 1 of length 3, 1 of length 4, 3 of length 6, and the improper subgroup consisting of the entire group. Since C_2×C_6 is Abelian, all its subgroups are normal. Since it has normal subgroups other than the trivial subgroup and the improper subgroup, C_2×C_6 is not a simple group.

See also

Cyclic Group C2, Cyclic Group C6

Explore with Wolfram|Alpha

Cite this as:

Weisstein, Eric W. "Finite Group C_2×C_6." From MathWorld--A Wolfram Web Resource.

Subject classifications