  We study permutation orbifolds of the $2$-fold and $3$-fold tensor product
for the Virasoro vertex algebra $\mathcal{V}_c$ of central charge $c$. In
particular, we show that for all but finitely many central charges
$\left(\mathcal{V}_c^{\otimes 3}\right)^{\mathbb{Z}_3}$ is a $W$-algebra of
type $(2, 4, 5, 6^3 , 7, 8^3 , 9^3 , 10^2 )$. We also study orbifolds of their
simple quotients and obtain new realizations of certain rational affine
$W$-algebras associated to a principal nilpotent element. Further analysis of
permutation orbifolds of the celebrated $(2,5)$-minimal vertex algebra
$\mathcal{L}_{-\frac{22}{5}}$ is presented.
