$D_{6}$ is called the dihedral group of symmetries of an equilateral triangle. Let $\rho_{0}, \rho_{1}, \rho_{2}$ denote anticlockwise rotations about $O$ through angles $0, \frac{2\pi}{3}, \frac{4\pi}{3}$ respectively. Let $\sigma_{1}, \sigma_{2}, \sigma_{3}$ denote reflections about the three different [[Altitude of Triangle|altitudes]]. ![[Symmetries of an Equilateral Triangle.png|700]] **Cayley Table** | | | $\rho_{0}$ | $\rho_{1}$ | $\rho_{2}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | | --------- | ------------ | ------------ | :----------: | ------------ | ------------ | ------------ | ------------ | | | $\rho_{0}$ | $\rho_{0}$ | $\rho_{1}$ | $\rho_{2}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | | | $\rho_{1}$ | $\rho_{1}$ | $\rho_{2}$ | $\rho_{0}$ | $\sigma_{3}$ | $\sigma_{1}$ | $\sigma_{2}$ | | Do second | $\rho_{2}$ | $\rho_{2}$ | $\rho_{0}$ | $\rho_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\sigma_{1}$ | | | $\sigma_{1}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\rho_{0}$ | $\rho_{1}$ | $\rho_{2}$ | | | $\sigma_{2}$ | $\sigma_{2}$ | $\sigma_{3}$ | $\sigma_{1}$ | $\rho_{2}$ | $\rho_{0}$ | $\rho_{1}$ | | | $\sigma_{3}$ | $\sigma_{3}$ | $\sigma_{1}$ | $\sigma_{2}$ | $\rho_{1}$ | $\rho_{2}$ | $\rho_{0}$ | ### Theorems - [[D6 is the smallest non-abelian group]]