 Solution

Choose transverse directions parallel and perpendicular to the vertical mirror plane. At each metal the parallel p orbital is unchanged, whereas the perpendicular p orbital changes sign. Thus each two-dimensional block has trace $1-1=0$. The total character of a representation is $\chi ({\sigma }_d)=0$. A mirror plane passing through an orbital centre does not imply that every orbital on that centre contributes $+1$.