Systoles of hyperbolic surfaces with big cyclic symmetry

We obtain the exact values of the systoles of these hyperbolic surfaces of genus g with cyclic symmetry of the maximum order and the next maximum order. Precisely, for the genus g hyperbolic surface with order 4 g + 2 cyclic symmetry, the systole is ( 1 + cosπ2g + 1 + cos2π2g + 1) when g ⩾ 7, and for the genus g hyperbolic surface with order 4 g cyclic symmetry, the systole is ( 1 + 2cosπ2g) when g ⩾ 4.