Past exam of the mathematics course of the year 2 of the natural sciences course of the University of Cambridge / 2022 / Paper 2 / 8 / c / Solution

Each reflection has order two, giving exactly the three order-two subgroupsConjugation by $r$ sends $r^{j}s$ to $r^{j+2}s$, cycling the three reflections. None of these subgroups is therefore normal.

The two nonidentity rotations generate the same order-three subgroup:It is invariant under conjugation by $r$, and $srs=r^{-1}$ shows invariance under $s$ as well; since $r,s$ generate the group, this proves normality. The complete partition into conjugacy classes isThe rotations are mutually conjugate by $s$, and the reflections by $r$; no rotation is conjugate to a reflection because conjugation preserves the order of a group element.