  We explore the feasibility of the discrete flavor symmetry $S_3$ to explain
the pattern of neutrino masses and mixings. In the flavor basis, the neutrino
mass matrix is taken to be invariant under $S_3$ symmetry at the zeroth order.
The effects of breaking $S_3$ symmetry by matrices invariant under different
$S_2$ subgroups of $S_3$ are studied. The resulting retrocirculant perturbation
matrix leads to the perturbed $S_3$ neutrino mass matrix having a trimaximal
eigenvector as suggested by the solar neutrino data. It is found that after the
third order perturbation the neutrino mixings only depend on the perturbation
parameter and are consistent with the current experimental data leading to very
small deviations from tribimaximal mixing. These perturbations in the $S_3$
invariant neutrino mass matrix result in interesting interplay between the
solar and the reactor neutrino mixing angles. We also get a strongly suppressed
range of effective Majorana mass which lies well within the reach of the
ongoing experiments.
