 Symmetry group of a square

Take $r$ as a counterclockwise quarter-turn and $s$ as reflection across the horizontal axis. The eight symmetries of a square are $e,r,r^2,r^3,s,rs,r^{2}s,r^{3}s$. The last four are reflections across the horizontal axis, the diagonal $y=x$, the vertical axis and the diagonal $y=-x$, respectively. This is a dihedral group of order eight; the subscript convention $D_4$ counts polygon vertices, while some authors use $D_8$ for the same group.