  Having in mind their potential quantum physical applications, we classify all
geometric hyperplanes of the near hexagon that is a direct product of a line of
size three and the generalized quadrangle of order two. There are eight
different kinds of them, totalling to 1023 = 2^{10} - 1 = |PG(9, 2)|, and they
form two distinct families intricately related with the points and lines of the
Veldkamp space of the quadrangle in question.
