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.
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