Definition:Symmetry Group of Square

Group Example
Let $\mathcal S = ABCD$ be a square.

The various symmetry mappings of $\mathcal S$ 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 $D_4$.

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