Symmetric Group acting on $X \times X$

Question

The symmetric group $S_n$ acts on the set $X = \{1,\ldots,n\}$ and hence acts on $X \times X$ by $g(x,y) = (gx, gy)$. Determine the orbits of $S_n$ on $X \times X$.

Not sure how do I actually "determine" the orbits? Thanks in advance for your help.

Answer 1

Hint: Try breaking $X \times X$ into $A:=\{(x,y) \in X \times X:x=y \}$ and $B=\{(x,y) \in X \times X:x \neq y \}$.