The dihedral group on the pentagon

Imagine a sheet of paper cut into a pentagon. On the arbitrarily designated "FRONT" size, label the vertices "a", "b", "c", "d" and "e" in a clockwise direction. On the reverse ("BACK") side, label the vertices "α", "β", "γ", "δ" and "&epsilon", corresponding to "a", "b", "c", "d" & "e", respectively. This labeling is counter-clockwise. The dihedral group of the pentagon is the group of all symmetries of the pentagon. This includes rotations through multiples of 2π/5 degrees, reflections about five different axes and flips about five different axes. Axisd, for example, is the line through vertex "d" that intersects the side of the pentagon opposite to it at right angles. In this case, the side connects vertices "a" and "b". Define ρ   Clockwise rotation through 2π/5 of the "FRONT" side, which is a counter-clockwise rotation through the same angle of the "BACK" side. τ   A flip about axisa. It turns out that ρ & τ generate the dihedral group of the pentagon. The two "radio buttons" to the right correspond to ρ & τ   ρ       τ                 |       |       FRONT         |                       |       |       BACK         |           List of actions: