How Can Orbits Depend on Each Other in Colored Graph Automorphisms?

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Hi,
I'm struggling with the following:
I have a colored graph, and I used Bliss to find the generators group for the automorphisms. Finding the orbits given the generators is easy.
What I'm trying to find now is dependence between orbits - for each orbit I want to know if it still exist if I"ll stabilize the rest of the graph, and if not, which of the other orbits need to be "free" to move with it. For example, if the generators are (1 2 3), (3 4)(6 7) and (5 8 9) then I have the orbits: (1 2 3 4), (6 7) and (5 8 9), however, (1 2 3 4) and (6 7) have to move together (so they are dependent) and (5 8 9) can move by itself (independently of the others).

I tried searching in the computational group theory literature without success, but I may be missing some terminology here, any help will be appreciated.

Thanks!
 
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This sounds as if you were in need of possible decomposition series of the automorphism group, i.e. its structure. E.g. if it was simple, then it wouldn't have any normal subgroups, which I assume correspond to subsets of fixed elements, but I'm not sure. Anyway, I would go in this direction.
 

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