Sliding piece puzzles with oriented tiles (Q1377741)

Let \(\theta_0\) be the hexagon with central vertex \(z\) adjacent with some pair of antipodal vertices. Theorem. The group of a sliding piece puzzle with oriented tiles on a nonseparable graph \({\mathcal G}\) is isomorphic to either the cyclic group of order \(n| H_0|\), if \({\mathcal G}\) is a polygon, or subgroup of the wreath product \(H_0\text{ wr }S_n\) of index \(i\) in the contrary case, \(i= 1,2,6\) or 12, with 12 possible only for \(\theta_0\).