Subgroups of Symmetry Group of Regular Hexagon

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

Let $D_6$ denote the symmetry group of $\mathcal H$.


 * SymmetryGroupRegularHexagon.png

Let $e$ denote the identity mapping

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

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

The subsets of $D_6$ which form its subgroups are as follows:


 * Order $1$:


 * $\set e$


 * Order $2$:


 * $\set {e, \alpha^3}$


 * $\set {e, \beta}$


 * $\set {e, \alpha \beta}$


 * $\set {e, \alpha^2 \beta}$


 * $\set {e, \alpha^3 \beta}$


 * $\set {e, \alpha^4 \beta}$


 * $\set {e, \alpha^5 \beta}$


 * Order $3$:


 * $\set {e, \alpha^2, \alpha^4}$


 * Order $4$:


 * $\set {e, \alpha^3, \beta, \alpha^3 \beta}$


 * $\set {e, \alpha^3, \alpha \beta, \alpha^4 \beta}$


 * $\set {e, \alpha^3, \alpha^2 \beta, \alpha^5 \beta}$


 * Order $6$:


 * $\set {e, \alpha, \alpha^2, \alpha^3, \alpha^4, \alpha^5}$


 * $\set {e, \alpha^2, \alpha^4, \beta, \alpha^2 \beta, \alpha^4 \beta}$


 * $\set {e, \alpha^2, \alpha^4, \alpha \beta, \alpha^3 \beta, \alpha^5 \beta}$


 * Order $12$:


 * $D_6$ itself.