Definition:Symmetry Group of Square

Group Example
Let $\mathcal R = ABCD$ be a (non-square) rectangle.

The various symmetry mappings of $\mathcal R$ 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 $t_{1,3}$ is a reflection about the diagonal through vertices $1$ and $3$ respectively.
 * The reflection $t_{2,4}$ is a reflection about the diagonal through vertices $2$ and $4$ respectively.

The symmetries of $\mathcal R$ form the dihedral group $D_4$.

Its Cayley table can be written:


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

& e & r & r^2 & r^3 & t_x & t_y & t_{1,3} & t_{2,4} \\ \hline e & e & r & r^2 & r^3 & t_x & t_y & t_{1,3} & t_{2,4} \\ r & r & r^2 & r^3 & e & t_{1,3} & t_{2,4} & t_x & t_y \\ r^2 & r^2 & r^3 & e & r & t_y & t_x & t_{2,4} & t_{1,3} \\ r^3 & r^3 & e & r & r^2 & t_{2,4} & t_{1,3} & t_x & t_y \\ t_x & t_x & t_{2,4} & t_y & t_{1,3} & e & r^2 & r^3 & r \\ t_y & t_y & t_{1,3} & t_x & t_{2, 4} & r^2 & e & r & r^3 \\ t_{1,3} & t_{1,3} & t_x & t_{2,4} & t_y & r & r^3 & e & r^2 \\ t_{2,4} & t_{2,4} & t_y & t_{1,3} & t_x & r^3 & r & r^2 & e\\ \end{array}$

Notation
Some sources denote $D_4$ as ${D_4}^*$.