Symmetry Group of a Regular Hexagon

The symmetry group of a regular hexagon is a group of order 12, the Dihedral group D6.

It is generated by a rotation R1 and a reflection r0. Rn denotes the rotation by angle n * 2 pi/6 with respect the center of the hexagon. rn denotes the reflection in the line at angle n * pi/6 with respect to a fixed line passing through the center and one vertex of the hexagon. Take the fixed line to be horizontal in the hexagon:

Then the effects of the elements of the group are as follows:

 id R1 R2 R3 R4 R5 r0 r1 r2 r3 r4 r5

The Cayley table is given below. An entry in a row labeled r and column labeled c represents the function composition r(c( )), thus c is applied first to the hexagon. Some people would say that this is backwards and that the conventional notation for function composition is very unfortunate. I'm inclined to agree. The function composition should be (( )r)c and thus the entry on the left, the row, should be the first element applied, in which case the transpose of the following table should be used.

 id R1 R2 R3 R4 R5 r0 r1 r2 r3 r4 r5 id id R1 R2 R3 R4 R5 r0 r1 r2 r3 r4 r5 R1 R1 R2 R3 R4 R5 id r1 r2 r3 r4 r5 r0 R2 R2 R3 R4 R5 id R1 r2 r3 r4 r5 r0 r1 R3 R3 R4 R5 id R1 R2 r3 r4 r5 r0 r1 r2 R4 R4 R5 id R1 R2 R3 r4 r5 r0 r1 r2 r3 R5 R5 id R1 R2 R3 R4 r5 r0 r1 r2 r3 r4 r0 r0 r5 r4 r3 r2 r1 id R5 R4 R3 R2 R1 r1 r1 r0 r5 r4 r3 r2 R1 id R5 R4 R3 R2 r2 r2 r1 r0 r5 r4 r3 R2 R1 id R5 R4 R3 r3 r3 r2 r1 r0 r5 r4 R3 R2 R1 id R5 R4 r4 r4 r3 r2 r1 r0 r5 R4 R3 R2 R1 id R5 r5 r5 r4 r3 r2 r1 r0 R5 R4 R3 R2 R1 id

The orders of the elements are as follows:

 Element: id R1 R2 R3 R4 R5 r0 r1 r2 r3 r4 r5 Order: 1 6 3 2 3 6 2 2 2 2 2 2

The subgroups are as follows:

 Elements: id R1 R2 R3 R4 R5 r0 r1 r2 r3 r4 r5 Subgroups: Trivial id C2 id R3 C3 id R2 R4 C6 id R1 R2 R3 R4 R5 C2 id r0 C2 id r1 C2 id r2 C2 id r3 C2 id r4 C2 id r5 C2+C2 id R3 r0 r3 C2+C2 id R3 r1 r4 C2+C2 id R3 r2 r5 D3 id R2 R4 r0 r2 r4 D3 id R2 R4 r1 r3 r5 Full D6 id R1 R2 R3 R4 R5 r0 r1 r2 r3 r4 r5