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# Dihedral Group D_6

The dihedral group gives the group of symmetries of a regular hexagon. The group generators are given by a counterclockwise rotation through radians and reflection in a line joining the midpoints of two opposite edges. If denotes rotation and reflection, we have

 (1)

From this, the group elements can be listed as

 (2)

The conjugacy classes of are given by

 (3)

The set of elements which by themselves make up conjugacy classes are in the center of , denoted , so

 (4)

The commutator subgroup is given by

 (5)

which can be used to find the Abelianization. The set of all left cosets of is given by

 (6) (7)

Thus we appear to have two generators for this group, namely and . Therefore, Abelianization gives .

It is also known that where is the symmetric group. Furthermore where is the dihedral group with 6 elements, i.e., the group of symmetries of an equilateral triangle.

There are thus two ways to produce the character table, either inducing from and using the orthogonality relations or simply by finding the character tables for and and taking their group direct sum.

 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 2 1 0 0 2 2 0 0

Dihedral Group

This entry contributed by Declan Davis

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## References

Arfken, G. Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 248, 1985.

## Cite this as:

Davis, Declan. "Dihedral Group D_6." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/DihedralGroupD6.html