 Solution

Use the ordered basis $(p_x,p_y,p_z)$ at the origin. Choose $z$ as the principal improper-rotation axis, $x,y$ along the two $C_2^{\prime }$ axes, and the diagonal mirror planes as $x=\pm y$. Use active operations with $\hat{U}(g)f(\mathbf{r})=f(g^{-1}\mathbf{r})$, so the matrix columns give the transformed orbitals. For these real p orbitals the matrices are the ordinary vector transformation matrices.

One representative from each D2d molecular point group class isFor example $S_4^{+}$ sends $p_x\mapsto p_y$, $p_y\mapsto -p_x$ and $p_z\mapsto -p_z$. Choosing the opposite sense of $S_4$ transposes its xy block but leaves every character and the final decomposition unchanged.