The
Dihedral group D_{n} is the
symmetry group
of the regular
ngon
^{1}.
This group has 2n elements. They are the rotations
given by the powers of r, rotation anticlockwise through
2pi/n, and the n reflections given by reflection
in the line through a vertex (or the midpoint of an edge) and the centre of the polygon.

\  /
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 /  \ 
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/  \
/  \
Let
s denote one of these reflections.
It's not hard to see that
D_{n}={1,r,r^{2},...,r^{n1},s,rs,...,r^{n1}s}
and that we have the relations:
r^{n}=1 s^{2}=1 and
srs^{1}=r^{1}. In fact the
Dihedral group is given by these
generators and
relations.
If we label the vertices of the polygon by {1,2,...,n}
then each symmetry of the polygon gives rise to a permutation
of {1,2...,n}. This gives a homomorphism to the
Symmetric group D_{n}>S_{n}.
It' s easy to see that it is an injection.
In the case n=4 this is what we get.
2 1

 
 

 
 

3 4
Let
s be
reflection in the
xaxis and
let
r be as before. Then
s corresponds
to the permutation
(23)(14) and
t corresponds
to (1234). Thus D_{4} is a
Sylow 2subgroup of S_{4}.
1. There is some disagreement about notation here. Some
people write D_{2n} for what I've called D_{n}