Describe the orbitals of transitive $G$ on $\Omega$

Question

Suppose $G\lt $ Sym$(\Omega)$ and $G$ acts transitively on $\Omega$ and has rank $3$.

I guess the orbitals may be the diagonal, and two other orbitals with same cardinalities forming from equally splitting the set $\{(x,y)^G|x\ne y, x\in\Omega, y\in\Omega\}$.

But I have no idea how to prove it. Can someone give me a hint or help?

Thanks a lot.

Update: As @anon pointed out, my guess is wrong. But we deduce that there are only two possible cases for the non-diagonal orbitals. One case is my guess, and the other case corresponds to the example given by @anon.

Answer 1

Your guess is wrong.

Counterexample: consider the symmetry group $G$ of a tetrahedron (note $G\cong S_4$) acting on the edge set $E$ of size $6$. Then $E^2$ has three orbits: the diagonal pairs (there are $6$ of these), the adjacent pairs (there are $24$ of these) and the disjoint pairs (there are $6$ of these). Note the nondiagonal elements of $E^2$ come in two orbits of different sizes.