Fill the $n\times n$ lattice square with natural numbers $\{1,2,\cdots,n\}$ each of which is used exactly $n$ times. Two configurations are in an equivalence class if and only if one is transformed from the other by any of the following group actions.
- A permutation of the numbers.
- Rotations of integer multiples of $\frac\pi 2$ about the axis perpendicular to the plane the square resides on.
- Reflection about vertical, horizontal and diagonal axises of the square.
How many equivalence classes of the configurations are there? What if we roll the square into a torus?
Do I need to apply Burnside's lemma?
To start with, $n=3$.