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Cyclic Group C_6


C_6 is one of the two groups of group order 6 which, unlike D_3, is Abelian. It is also a cyclic. It is isomorphic to C_2×C_3. Examples include the point groups C_6 and S_6, the integers modulo 6 under addition (Z_6), and the modulo multiplication groups M_7, M_9, and M_(14) (with no others).

CyclicGroupC6CycleGraph

The cycle graph is shown above and has cycle index

 Z(C_6)=1/6x_1^6+1/6x_2^3+1/3x_3^2+1/3x_6.

The elements A_i of the group satisfy A_i^6=1, where 1 is the identity element, three elements satisfy A_i^3=1, and two elements satisfy A_i^2=1.

CyclicGroupC6Table

Its multiplication table is illustrated above and enumerated below.

C_61ABCDE
11ABCDE
AABCDE1
BBCDE1A
CCDE1AB
DDE1ABC
EE1ABCD

Since C_6 is Abelian, the conjugacy classes are {1}, {A}, {B}, {C}, {D}, and {E}. There are four subgroups of C_6: {1}, {1,C}, {1,B,D}, and {1,A,B,C,D,E} 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 C7, Cyclic Group C8, Cyclic Group C9, Cyclic Group C10, Cyclic Group C11, Cyclic Group C12, Dihedral Group D3

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Cite this as:

Weisstein, Eric W. "Cyclic Group C_6." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/CyclicGroupC6.html

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