An F_p²-maximal Wiman sextic and its automorphisms

In 1895 Wiman introduced the Riemann surface W of genus 6 over the complex field C defined by the equation X6+Y6+Z6+(X2+Y2+Z2)(X4+Y4+Z4)-12X2Y2Z2 = 0, and showed that its full automorphism group is isomorphic to the symmetric group S5. We show that this holds also over every algebraically closed field Kof characteristic p ≥ 7.

For p = 2, 3 the above polynomial is reducible over K, and for p = 5 the curve W is rational and Aut(W) ≅ PGL(2,K). We also show that Wiman's F192-maximal sextic W is not Galois covered by the Hermitian curve H19 over the finite field H192.