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# Dihedral Group

The dihedral group is the symmetry group of an -sided regular polygon for . The group order of is . Dihedral groups are non-Abelian permutation groups for .

The th dihedral group is represented in the Wolfram Language as DihedralGroup[n].

One group presentation for the dihedral group is .

A reducible two-dimensional representation of using real matrices has generators given by and , where is a rotation by radians about an axis passing through the center of a regular -gon and one of its vertices and is a rotation by about the center of the -gon (Arfken 1985, p. 250).

Dihedral groups all have the same multiplication table structure. The table for is illustrated above.

The cycle index (in variables , ..., ) for the dihedral group is given by

 (1)

where

 (2)

is the cycle index for the cyclic group , means divides , and is the totient function (Harary 1994, p. 184). The cycle indices for the first few are

 (3) (4) (5) (6) (7)

Renteln and Dundes (2005) give the following (bad) mathematical joke about the dihedral group:

Q: What's hot, chunky, and acts on a polygon? A: Dihedral soup.

## See also

Dihedral Group D3, Dihedral Group D4, Dihedral Group D5, Dihedral Group D6 Explore this topic in the MathWorld classroom

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## References

Arfken, G. "Dihedral Groups, ." Mathematical Methods for Physicists, 3rd ed. Orlando, FL: Academic Press, p. 248, 1985.Harary, F. Graph Theory. Reading, MA: Addison-Wesley, pp. 181 and 184, 1994.Lomont, J. S. "Dihedral Groups." §3.10.B in Applications of Finite Groups. New York: Dover, pp. 78-80, 1987.Renteln, P. and Dundes, A. "Foolproof: A Sampling of Mathematical Folk Humor." Notices Amer. Math. Soc. 52, 24-34, 2005.

Dihedral Group

## Cite this as:

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