Symmetry Group of Regular Hexagon/Examples/Subgroup Generated by alpha^4 and alpha^3 beta

Examples of Operations on Symmetry Group of Regular Hexagon
Let $\HH = ABCDEF$ be a regular hexagon.

Let $D_6$ denote the symmetry group of $\HH$.


 * SymmetryGroupRegularHexagon.png

Let $e$ denote the identity mapping

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

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

Let $H$ be the subgroup of $D_6$ generated by $\alpha^4$ and $\alpha^3 \beta$.

Then:
 * $H = \set {e, \alpha^2, \alpha^4, \alpha \beta, \alpha^3 \beta, \alpha^5 \beta}$

and:
 * $H \cong D_3$

Proof
Demonstration by Cayley table:


 * $\begin{array}{c|cccccc}

& e & \alpha^2 & \alpha^4 & \alpha \beta & \alpha^3 \beta & \alpha^5 \beta \\ \hline e             & e              & \alpha^2       & \alpha^4       & \alpha \beta   & \alpha^3 \beta & \alpha^5 \beta \\ \alpha^2      & \alpha^2       & \alpha^4       & e              & \alpha^3 \beta & \alpha^5 \beta & \alpha \beta   \\ \alpha^4      & \alpha^4       & e              & \alpha^2       & \alpha^5 \beta & \alpha \beta   & \alpha^3 \beta \\ \alpha  \beta & \alpha   \beta & \alpha^5 \beta & \alpha^3 \beta & e              & \alpha^4       & \alpha^2       \\ \alpha^3 \beta & \alpha^3 \beta & \alpha \beta  & \alpha^5 \beta & \alpha^2       & e              & \alpha^4       \\ \alpha^5 \beta & \alpha^5 \beta & \alpha^3 \beta & \alpha \beta  & \alpha^4       & \alpha^2       & e \\ \end{array}$

It can be seen by inspection that this is $D_3$.