Past exam of the chemistry course of the year 3 of the natural sciences course of the University of Cambridge / 2022 / Paper 2B / Section A / 28 / b / Solution

A $180^{\circ }$ rotation leaves both metal centres fixed but reverses each transverse p orbital: $p_x\mapsto -p_x$ and $p_y\mapsto -p_y$. Its matrix is $-I_4$, so its character of a representation is $\chi (C_6^3)=-4$, also obtained from $4\cos \pi$.