Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 156 / 1 / d / Solution

Take the genus-two surface obtained from a regular hyperbolic octagon fundamental polygon by identifying opposite sides. Rotation of the octagon through $\pi /4$ respects the side pairing and descends to an orientation-preserving isometry of order eight. Thus $G=C_8$ acts on $S_2$ andThe fixed image of the octagon centre explains why this does not contradict part c.