  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.
