Definition:Symmetry Group of Regular Hexagon

Group Example
Let $\mathcal H = ABCDEF$ be a regular hexagon.


 * SymmetryGroupRegularHexagon.png

The various symmetry mappings of $\mathcal H$ are:
 * The identity mapping $e$
 * The rotations through multiples of $60^\circ$
 * The reflections in the indicated axes.

Let $\alpha$ denote rotation of $\mathcal H$ anticlockwise through $\dfrac \pi 3$ radians ($60^\circ$).

Let $\beta$ denote reflection of $\mathcal H$ in the $AD$ axis.

The symmetries of $\mathcal H$ form the dihedral group $D_6$.

$D_6$ acts on the vertices of $\mathcal H$ according to this table:


 * $\begin{array}{cccccccccccc}

e & \alpha & \alpha^2 & \alpha^3 & \alpha^4 & \alpha^5 & \beta & \alpha \beta & \alpha^2 \beta & \alpha^3 \beta & \alpha^4 \beta & \alpha^5 \beta \\ \hline A & B & C & D & E & F & A & B & C & D & E & F \\ B & C & D & E & F & A & F & A & B & C & D & E \\ C & D & E & F & A & B & E & F & A & B & C & D \\ D & E & F & A & B & C & D & E & F & A & B & C \\ E & F & A & B & C & D & C & D & E & F & A & B \\ F & A & B & C & D & E & B & C & D & E & F & A \\ \end{array}$