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.
."