### Reflections in the symmetry group 2H_6 of a hexagon

The group 2H_6 has twelve distinct (oriented) reflections x -> x' = -rxr, rr=1. Each reflection is characterized by the unit direction vector r of the line of reflection. These reflections are oriented, because each line of reflections has two possible orientations, characterized by the sign of the vector r.
We have:
r = a, aba, a(ba)(ba) ... blue
r= -a, -aba, -a(ba)(ba) ... light blue
r= b, bab, b(ab)(ab) ... light green
r= -b, -bab, -b(ab)(ab) ... green.

Each reflection can be visualized by dragging the red direction vector r of the line of reflection to a particular value of r. It is easily verified that other choices of r, not in the above list, will not produce a symmetry of the hexagon. The combination of two such reflections at r and r' will yield an (oriented) rotation from the dicyclic group 2C_6.

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Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).