The dihedral group is the symmetry group of an sided regular polygon for . The group order of is . Dihedral groups are nonAbelian 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 twodimensional 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.