Dihedral Group

The dihedral group D_n is the symmetry group of an n-sided regular polygon for n>1. The group order of D_n is 2n. Dihedral groups D_n are non-Abelian permutation groups for n>2.

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

One group presentation for the dihedral group D_n is <x,y|x^2=1,y^n=1,(xy)^2=1>.

A reducible two-dimensional representation of D_n using real matrices has generators given by S and R, where S is a rotation by pi radians about an axis passing through the center of a regular n-gon and one of its vertices and R is a rotation by 2pi/n about the center of the n-gon (Arfken 1985, p. 250).


Dihedral groups all have the same multiplication table structure. The table for D_(10) is illustrated above.

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

 Z(D_p)=1/2Z(C_p)+{1/2a_1a_2^((p-1)/2)   for p odd; 1/4(a_2^(p/2)+a_1^2a_2^((p-2)/2))   for p even,



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


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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Arfken, G. "Dihedral Groups, D_n." 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.

Referenced on Wolfram|Alpha

Dihedral Group

Cite this as:

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

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