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:


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 idR1R2R3R4R5r0r1r2r3r4r5
R1 R1R2R3R4R5idr1r2r3r4r5r0
R2 R2R3R4R5idR1r2r3r4r5r0r1
R3 R3R4R5idR1R2r3r4r5r0r1r2
R4 R4R5idR1R2R3r4r5r0r1r2r3
R5 R5idR1R2R3R4r5r0r1r2r3r4
r0 r0r5r4r3r2r1idR5R4R3R2R1
r1 r1r0r5r4r3r2R1idR5R4R3R2
r2 r2r1r0r5r4r3R2R1idR5R4R3
r3 r3r2r1r0r5r4R3R2R1idR5R4
r4 r4r3r2r1r0r5R4R3R2R1idR5
r5 r5r4r3r2r1r0R5R4R3R2R1id

The orders of the elements are as follows:

Element: idR1R2R3R4R5r0r1r2r3r4r5
Order: 163236222222

The subgroups are as follows:

Elements: idR1R2R3R4R5r0r1r2r3r4r5
Trivial id
C2 idR3
C3 idR2R4
C6 idR1R2R3R4R5
C2 idr0
C2 idr1
C2 idr2
C2 idr3
C2 idr4
C2 idr5
C2+C2 idR3r0r3
C2+C2 idR3r1r4
C2+C2 idR3r2r5
D3 idR2R4r0r2r4
D3 idR2R4r1r3r5
Full D6 idR1R2R3R4R5r0r1r2r3r4r5