  In this work, we analyze the Higgs sector of the minimal $S(3)$-invariant
extension of the Standard Model is performed. Considering three Higgs fields,
which are SU(2) doublets, and CP invariant, we compute the exact and analytical
physical Higgs boson masses in terms of the Higgs potential parameters and the
scalar Higgs matrix rotation angle $\theta_S$ and $w_3$
($\tan\theta_P=\tan\theta_C=\tan^{-1}\omega_3$), related to the pseudoscalar
and charged Higgs matrix rotation angles $\theta_P$ and $\theta_C$
respectively. Furthermore, within this model we can also write down in an
explicit form the trilinear self-couplings $\lambda_{ijk}$ in terms of the
Higgs masses and two free parameters, $\theta_S$ and $w_3$. Moreover, we show
that the Higgs masses and trilinear Higgs bosons self-couplings are closely
linked to the Higgs potential structure given by the discrete symmetry $S(3)$,
which can be helpful to distinguish this model from other extensions. In our
analysis the lightest Higgs boson mass is taken to be fixed to 125 GeV. In
concordance with the results reported in the literature for other Standard
Model extensions, it find that the numerical values $\lambda_{ijk}$ of the
minimal $S(3)$-invariant extension of the Standard Model are significantly
different from the trilinear Higgs self-coupling of the Standard Model.
