  In this work, we delve into the often-overlooked cosmological implications of
spontaneous breaking of non-Abelian discrete groups, specifically focusing on
the formation of domain walls in the case of $S_{4}$ flavor symmetry. In
particular, we investigate three interesting breaking patterns of the $S_4$
group and study the structure of the domain walls in the broken phase for three
possible residual symmetries. The presentation of domain walls in the case of
multiple vacua is usually complicated, which therefore implies that most of the
analyzes only approximate their presentation. Here, we propose a subtle way to
represent the $S_{4}$ domain wall networks by presenting the vacua in each
breaking pattern as vectors with their components corresponding to their
coordinates in the flavon space. Then, through the properties of the obtained
vectors, we find that the domain wall networks can be represented by Platonic
or Archimedean solids where the vertices represent the degenerate vacua while
the edges correspond to the domain walls that separate them. As an
illustration, we propose a type-II seesaw model that is based on the $S_{4}$
flavor symmetry, and study its phenomenological implications on the neutrino
sector. To solve the domain wall problem within this toy model, we consider an
approach based on high-dimensional effective operators induced by gravity that
explicitly break the structure of the induced vacua favoring one vacuum over
the others.
