Minimal block system in GAP software

Question

I want to list down all minimal block system, I am trying "Blocks(G,[1..4]);" command but it is not giving minimal block system. Let $G=\langle (12)(34),(13)(24),(14)(23) \rangle$ $$\pi: G \times [4] \mapsto [4]$$

Minimal block system : First consider the group action ($\phi$) of $G$ on $[n]$, then we will be get the blocks , so we need to refine $G$ means at the second level from the leafs, we will consider the action of group $G/Ker(\phi)$ and so on.

Is there any inbuilt command to print all minimal block systems? Thank you.

Answer 1

Try RepresentativesMinimalBlocks(G,MovedPoints(G)). It will give for each block system the block containing $1$. If $r$ is such a block Orbit(g,Set(r),OnSets) will return the full block system.