The group 2C_6 has six rotations x->x'= R~xR (aa=1, bb=1) with "positive" sense represented by the rotors R:
1=(ab)^0, ab, (ab)^2, (ab)^3, (ab)^4, (ab)^5
and six corresponding rotations with "negative" sense represented by the rotors R:
ba=-(ab)^5, (ba)^2=-(ab)^4, (ba)^3=-(ab)^3, (ba)^4=-(ab)^2, (ba)^5=-ab, (ba)^6=(ab)^6=-1.
Here the rotations are realized by double reflections x->x' -- similar to the applet Rotations by reflection. The results after the first reflection (at the blue line through a) are not shown explicitly. The final results x' after the second reflection (at the red line of reflection) are shown as rings of the corresponding color marked by numbers with a prime. One can "see" all the above 12 rotations by interactively changing the position of the bright red point (this changes the second red line of reflection). We get symmetries only when the red vector coincides with one of the six green (including a and b) [positive] vectors or one of the six orange [negative] vectors. These symmetry operations are in 1:1 correspondence with the rotors listed above. The single reflections, which pairwise yield the rotations x->x' are members of the symmetry group of the hexagon.
Soli Deo Gloria. Created with Cinderella by Eckhard Hitzer (Fukui).