  In this paper, a continuation of \cite{MPS}, we investigate the
$S_3$-orbifold subalgebra of $(\mathcal{V}_c)^{\otimes 3}$, that is, we
consider the $S_3$-fixed point vertex subalgebra of the tensor product of three
copies of the universal Virasoro vertex operator algebras $\mathcal{V}_c$. Our
main result is construction of a minimal, strong set of generators of this
subalgebra for any generic values of $c$. More precisely, we show that this
vertex algebra is of type $(2,4,6^2,8^2,9,10^2,11,12^3)$.
  We also investigate two prominent examples of simple $S_3$-orbifold algebras
corresponding to central charges $c=\frac12$ (Ising model) and
$c=-\frac{22}{5}$ (i.e. $(2,5)$-minimal model). We prove that the former is a
new unitary $W$-algebra of type $(2,4,6,8)$ and the latter is isomorphic to the
affine simple $W$-algebra of type $\frak{g}_2$ at non-admissible level
$-\frac{19}{6}$. We also provide another version of this isomorphism using the
affine $W$-algebra of type $\frak{g}_2$ coming from a subregular nilpotent
element.
