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Consider a cubic surface satisfying the mild condition that it may bedescribed in Sylvester's pentahedral form. There is a well-known Enriques orCoble surface S with K3 cover birationally isomorphic to the Hessian surface ofthis cubic surface. We describe the nef cone and the (-2)-curves of S. In thecase of pentahedral parameters (1, 1, 1, 1, nonzero t) we compute theautomorphism group of S. For t not 1 it is the semidirect product of the freeproduct (Z/2)*(Z/2)*(Z/2)*(Z/2) by the symmetric group S4. In the special caset=1/16 we study the action of Aut(S) on an invariant smooth rational curve C onthe Coble surface S. We describe the action and its image, both geometricallyand arithmetically. In particular, we prove that Aut(S)-->Aut(C) is injectivein characteristic 0 and we identify its image with the subgroup of PGL2 comingfrom the symmetries of a regular tetrahedron and the reflections across itsfacets.

The tetrahedron and automorphisms of Enriques and Coble surfaces of Hessian type