  In this manuscript, we present an explicit way to describe the violation of
CP symmetry in the standard model (SM) of electroweak interactions. In such a
way, complex Cabbibo-Kobayashi-Maskawa (CKM) matrices are achieved which stand
for the violation of CP symmetry. At the beginning, two necessary but not
sufficient conditions for yielding a complex CKM matrix are stated as criteria.
Then we found an interesting condition between the real and imaginary parts of
a Hermitian $3\times 3$ matrix may provide extra relations among its parameters
and reduce the number of them from eighteen down to five. In previous
investigations, this can be done only down to nine. With another assumption
among some of those parameters, the mass-matrix pattern is further simplified
so as to be diagonalized analytically and in consequence four matrices which
reveal $S_N$ symmetries among or between quark generations are obtained. In
some of such $S_N$-symmetric cases, the derived CKM matrices are complex which
indicate that CP symmetry is violated accordingly. Taking the Jarlskog
invariant as an estimate of the CPV strength, the value predicted by this model
is orders stronger than the empirical value detected experimentally. However,
that happens to fill partly the gap between the cosmologically observed amount
of Baryon Asymmetry of the Universe (BAU) and that current Standard Model of
particle physics predicts. It also proves the long suspected existence of
BAU-productive eras in early universe if some fermions were indistinguishable,
i.e., $S_N$-symmetric, under circumstances of extremely high temperatures.
