Let $G$ be a group acting on a set $X$. Then we have a natural action on $X \times X$ in the following way: $g. (x,y) = (gx,gy)$. Then, suppose we have two points of interest $x_1,x_2 \in X$, and we know their orbits under $G$. Is there some procedure to help up calculate the orbit of $(x_1,x_2)$?
edit: for what it's worth, both $G$ and $X$ are finite.