Definition:Dihedral Group D6

Example of Dihedral Group

The dihedral group $D_6$ is the symmetry group of the regular hexagon:

Let $\HH = ABCDEF$ be a regular hexagon.

The various symmetry mappings of $\HH$ are:

The identity mapping $e$

The rotations through multiples of $60 \degrees$ anticlockwise about the center of $\HH$

The reflections in the indicated axes.

Let $\alpha$ denote rotation of $\HH$ anticlockwise through $\dfrac \pi 3$ radians ($60 \degrees$).

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

The symmetries of $\HH$ form the dihedral group $D_6$.

Matrix Representations

Dihedral Group D6/Matrix Representation