 TOPICS  # Finite Group C_2×C_6

The finite group 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 , , , and (and no other modulo multiplication groups). The multiplication table is illustrated above. The numbers of elements for which for , 2, ..., 12 are 1, 4, 3, 4, 1, 12, 1, 4, 3, 4, 1, 12.

Each element of 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 is Abelian, all its subgroups are normal. Since it has normal subgroups other than the trivial subgroup and the improper subgroup, is not a simple group.

Cyclic Group C2, Cyclic Group C6

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## Cite this as:

Weisstein, Eric W. "Finite Group C_2×C_6." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/FiniteGroupC2xC6.html