Symmetry group of a square ($D_4=⟨r,s\mid r^4=s^2=e,srs=r^{-1}⟩$)

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.

In the order of the conjugacy classes of the symmetry group of a square displayed there, its irreducible complex character table isThe four one-dimensional representations assign independent signs to $r$ and $s$. The remaining row is the character of the ordinary planar representation: a rotation has trace $2\cos \theta$ and a reflection has trace zero. Weighted by class sizes, these rows satisfy character orthogonality. Their squared dimensions sum to eight, so they exhaust the irreducible representations.

The displayed conjugacy classes have sizes one, one, two, two and two. The identity and half-turn are central elements. Reflection sends $r$ to $r^{-1}$, pairing the two quarter-turns. Conjugating a reflection by $r$ sends $r^{k}s$ to $r^{k+2}s$; conjugating by $s$ sends it to $r^{-k}s$. Thus the parity of $k$ is invariant under conjugation, separating the axis reflections from the diagonal reflections, and each displayed pair is one conjugacy class.