  It has been recently claimed that the symmetry group S4 yields to the
Tri-bimaximal neutrino mixing in a "natural" way from the group theory point of
view. Approving of this feature as an indication, we build a supersymmetric
model of lepton and quark masses based on this family symmetry group. In the
lepton sector, a correct mass hierarchy among the charged leptons is achieved
together to a neutrino mass matrix which can be diagonalized by the
Tri-bimaximal pattern. Our model results to be phenomenologically unequivalent
with respect to other proposals based on different flavour groups but still
predicting the Tri-bimaximal mixing. In the quark sector a realistic pattern
for masses and mixing angles is obtained. The flavour structures of the mass
matrices in both the sectors come from the spontaneously symmetry breaking of
S4, due to several scalar fields, which get non-zero vacuum expectation values.
A specific vacuum alignment is required and it is shown to be a natural results
of the minimization of the scalar potential and, moreover, to be stable under
the corrections from the higher order terms.
