The discriminant of a cubic surface

We construct explicit examples of cubic surfaces over $\bbQ$ such that the 27 lines are acted upon by the index two subgroup of the maximal possible Galois group. This is the simple group of order $25 920$. Our examples are given in pentahedral normal form with rational coefficients. For such cubic surfaces, we study the discriminant and show its relation to the index two subgroup. On the corresponding parameter space, we search for rational points, discuss their asymptotic, and construct an accumulating subvariety.