The Dihedral group Dn is the symmetry group of the regular n-gon1.

This group has 2n elements. They are the rotations given by the powers of r, rotation anti-clockwise 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.

```         |
\       |       /
\ ----|---- /
| \   |   / |
|   \ | /   |
------------------
|   / | \   |
| /   |   \ |
/ ----|---- \
/       |       \
```
Let s denote one of these reflections.

It's not hard to see that

```Dn={1,r,r2,...,rn-1,s,rs,...,rn-1s}
```
and that we have the relations: rn=1 s2=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 Dn->Sn. 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 x-axis and let r be as before. Then s corresponds to the permutation (23)(14) and t corresponds to (1234). Thus D4 is a Sylow 2-subgroup of S4.

1. There is some disagreement about notation here. Some people write D2n for what I've called Dn