How many representations of a non-commutative groups from $n$-dimensional Latin Squares

Question

I would like to know if there is a method to determine how many $n \times n$ Latin square define the same non-commutative group $G$. This is how many representations of $G$ can be obtained from $n$ dimensional Latin squares.

It results that I'm interested on building examples of non-abelian groups to toy with and test how many representations do they have.

Answer 1

(I'm going to assume that the elements of the Latin Square are the elements of the group. If that's not what you want, then make your question clearer.)

For a given group, once you've chosen how you label the rows and columns, the Cayley table is uniquely determined. So if the group has order $n$, it has $(n!)^2$ tables.