Cyclic Group C_12

The cyclic group C_(12) 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 m=13 and 26 (which are the only modulo multiplication groups isomorphic to C_(12)).


The cycle graph of C_(12) is shown above. The cycle index is


Its multiplication table is illustrated above.

The numbers of elements satisfying A^i=1 for i=1, 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: {1}, {1,F}, {1,D,H}, and {1,C,F,I}. {1,B,D,F,H,J}, and {1,A,B,C,D,E,F,G,H,I,J,K} which, because the group is Abelian, are all normal. Since C_6 has normal subgroups other than the trivial subgroup and the entire group, it is not a simple group.

See also

Cyclic Group, Cyclic Group C2, Cyclic Group C3, Cyclic Group C4, Cyclic Group C5, Cyclic Group C6, Cyclic Group C7, Cyclic Group C8, Cyclic Group C9, Cyclic Group C10, Cyclic Group C11, Dihedral Group D4, Finite Group C2×C6

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Cite this as:

Weisstein, Eric W. "Cyclic Group C_12." From MathWorld--A Wolfram Web Resource.

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