2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78585It is one of the wonderful ``coincidences'' of the theory of finite groups that the simple group G of order 25920 arises as both a symplectic group in characteristic 3 and a unitary group in characteristic 2. These two realizations of G yield two G-covers of the moduli space of configurations of six points on the projective line modulo PGL_2, via the 3- and 2-torsion of the Jacobians of the double and triple cyclic covers of P^1 branched at those six points. Remarkably these two covers are isomorphic. This was proved over C by transcendental methods by Hunt and Weintraub. We give an algebraic proof valid over any field not of characteristic 2 or 3 that contains the cube roots of unity. We then explore the connection between this $G$-cover and the elliptic surface $y^2 = x^3 + sextic(t), whose Mordell-Weil lattice is E_8 with automorphisms by a central extension of G.23 pagesAlgebraic GeometryNumber Theory14H10,14D22,14H45 (Primary), 14J25,14J26,14J27,14H40 (Secondary)text