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The dihedral group of order $2n$, $D_n$, acts faithfully on the set of the $n$ vertices of the regular $n$-gon. For $n=6$ and $n=10$, $D_n$ acts faithfully also on sets of size smaller than $n$, respectively $5$ and $7$.

What is the geometrical interpretation of these two faithful actions? In particular, what are the $5$ and $7$ "geometrical objects" of the regular hexagon and decagon, respectively, on which $D_6$ and $D_{10}$ act faithfully?

Edit. I think I got the point, driven by the comments: when $n=2m$ with $m$ odd, the two $m$-gon inscribed in the $n$-gon and the $m$ pairs of adjacent sides build up a set of size $m+2$ on which $D_n$ acts faithfully.

citadel
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    For $n=2m$ with $m$ odd, we have $D_n \cong D_m \times C_2$, and the actions that you are talking about are intransitive actions, with $D_m$ in its natural action on $m$ points, and the $C_2$ acting on the other two points - so the faithful action of $D_n$ is on $m+2$ points with orbits of sizes $m$ and $2$. – Derek Holt Mar 07 '23 at 22:02
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    A downvote and a vote to close?!? – Mariano Suárez-Álvarez Mar 07 '23 at 22:42
  • I don't quite get it. Why m needs to be odd then? – agent_cracker103 Mar 08 '23 at 22:50
  • This has been asked for $D_6$ before here (I think this is close enough to vote to close as a duplicate). You can find the more general construction on an old blog post of mine. – HallaSurvivor Mar 08 '23 at 22:59
  • @agent_cracker103, it needs for the action to be faithful. I've reworded the Edit to make it clearer – citadel Mar 09 '23 at 06:01
  • I see, @HallaSurvivor, thanks. Before acknowledging mine as duplicate, I wonder whether mine "pairs of adjacent sides" properly play the role of the $2$-gons (diagonals). – citadel Mar 09 '23 at 06:29

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