  We describe a supersymmetric Grand Unified Theory based on the gauge group
$\SU(5)^3$, or $\SO(10)^3$, invariant under the interchange of any~SU(5),
or~SO(10), with each family multiplet transforming non trivially under one
different individual group factor. A realistic pattern of fermion masses and
mixings is obtained as a result of an appropriate choice out of the many
possible discretely degenerate vacua of the theory. In the~SO(10) case, we
predict the three neutrino masses in terms of an overall scale and, within
factors of order unity, their mixing angles. A $\nu_e$-$\nu_\mu$ oscillation is
suggested as a solution of the solar neutrino problem, implying a visible
$\nu_\mu$-$\nu_\tau$ oscillation in the forthcoming experiments. Grand unified
theories of this type could be obtainable in a string theory framework.
