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)
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where
(2)
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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)
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(4)
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(5)
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(6)
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(7)
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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.