Transitive graph such that the stabilizer of a point has three orbits I am looking for an example of a finite graph such that its automorphism group is transitive on the set of vertices, but the stabilizer of a point has exactly three orbits on the set of vertices. I can't find such an example. Anyone has a suggestion?

Consider the Petersen graph. Its automorphism group is $S_5$ which acts on $2$-subsets of $\{1,\ldots,5\}$ (that can be seen as vertices of the graph). Now the result is clear.
