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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?

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    For the stabilizer's orbits: is one of the three orbits just the point being stabilized, or do you want three orbits other than that one?2011-08-15

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I believe a square has a vertex-transitive automorphism group, but a point stabilizer has three orbits (the point stabilizer, the opposite point, and the other two points).

I believe a heptagon has a vertex-transitive automorphism group, but a point stabilizer has three non-trivial orbits (and one additional trivial orbit consisting of the point stabilized).

(Square being a name for the cycle on 4 vertices with edge {12, 23, 34, 41}, and heptagon being a name for the cycle on 7 vertices with edges {12, 23, 34, 45, 56, 67, 71}.)

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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.