Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2023 / Paper 2 / 10 / c / Solution

The character of a representation is the trace of its representing matrix. For the rotation powers,Each $U^{j}V$ has a planar reflection block of trace zero and third diagonal entry $-1$, giving character $-1$. In the chosen element order,Equivalently its values on the conjugacy classes $\{I\}$, $\{U^2\}$, $\{U,U^3\}$, $\{V,U^{2}V\}$, $\{UV,U^{3}V\}$ are $3,-1,1,-1,-1$.