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Atkinson as well as Schönert and Seress describe methods to compute the minimal block system for transitive permutation groups; in particular in Permutation Group Algorithms by Ákos Seress, we find

Theorem 5.5.1 Suppose that a set S of generators for some transitive $G \leq Sym(\Omega)$ is given and $|\Omega| = n$. Then a minimal block of imprimitivity can be computed [...] by a deterministic algorithm.

Is there a way to compute such block systems for non-transitive permutation groups?

I have not found anything in the literature about such a computation.

Ingolfur
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    You just go through all orbits of $G$, one by one and compute block systems for each orbit. – ahulpke Jun 16 '22 at 07:24
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    I am once again mystified by the vote to close. – Derek Holt Jun 16 '22 at 08:28
  • @ahulpke, thank you for your reply. However, I am afraid I do not totally understand. – Ingolfur Jun 16 '22 at 08:48
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    I think you need to say exactly what it is that you do not understand. – Derek Holt Jun 16 '22 at 10:11
  • @DerekHolt, thank you. How does one compute block systems for each orbit? An orbit is just a set. – Ingolfur Jun 16 '22 at 10:45
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    The group acts transitively on each orbit and, as you said yourself in your post, you can use the method in Seress' book to find minimal blocks in that case. You could also use the earlier algorithm of Atkinson, which has worse complexity, but is easier to understand. – Derek Holt Jun 16 '22 at 11:19
  • @DerekHolt, thank you. I think this is the answer to my question. – Ingolfur Jun 16 '22 at 14:13

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