Symmetry Group of Square/Cayley Table

Cayley Table of Symmetry Group of Square
The Cayley table of the symmetry group of the square can be written:


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

& e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\ \hline e     & e & r & r^2 & r^3 & t_x & t_y & t_{AC} & t_{BD} \\ r     & r & r^2 & r^3 & e & t_{AC} & t_{BD} & t_y & t_x \\ r^2   & r^2 & r^3 & e & r & t_y & t_x & t_{BD} & t_{AC} \\ r^3   & r^3 & e & r & r^2 & t_{BD} & t_{AC} & t_x & t_y \\ t_x   & t_x & t_{BD} & t_y & t_{AC} & e & r^2 & r^3 & r \\ t_y   & t_y & t_{AC} & t_x & t_{BD} & r^2 & e & r & r^3 \\ t_{AC} & t_{AC} & t_x & t_{BD} & t_y & r & r^3 & e & r^2 \\ t_{BD} & t_{BD} & t_y & t_{AC} & t_x & r^3 & r & r^2 & e\\ \end{array}$ where the various symmetry mappings of the square $\mathcal S = ABCD$ are:
 * the identity mapping $e$
 * the rotations $r, r^2, r^3$ of $90^\circ, 180^\circ, 270^\circ$ counterclockwise respectively
 * the reflections $t_x$ and $t_y$ are reflections about the $x$ and $y$ axis respectively
 * the reflection $t_{AC}$ is a reflection about the diagonal through vertices $A$ and $C$
 * the reflection $t_{BD}$ is a reflection about the diagonal through vertices $B$ and $D$.