I was trying to use Burnside's formula on a regular hexagon. I believe the answer is $13$, but I am trying to show my work. Here is what I have figured out.

$$\frac{1}{12}(2^6 + 36 + 30 + \cdots?)$$

$2^6$ is $D\_{6}$ (at least that is what I think the regular hexagon is called). Then I reflected over the vertices. This gave me $3$ different possible reflections with $12$ different possible ways which is $36$. Then I reflected over the midpoints. This gave me $3$ different possible reflections with $10$ different possible ways which is $30$. Now the rotations are a bit confusing to me.