I would like to know exactly how many non-trivial blocks has $D_{16}$ of order $16$ acting on $8$ vertices of $8$-gon. I guess I should do it taking blocks as unions of the orbit of stabilisers but I don't know exactly how to do it.
2026-03-29 10:08:52.1774778932
Non-trivial blocks of order $16$ of $D_{16}$ acting on $8$-gon vertices
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Hint. The action of $D_{16}$ on the set $X$ of vertices is clearly transitive, so the blocks containing a fixed vertice $x\in X$ are in one-to-one correspondence with the subgroups of $D_{16}$ containing the stabilizer $G_x$ of $x$.
Now , what are the rotations fixing $x$ ? what are the symmetries fixing $x$?