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Let $p$ be an odd prime and let $\widehat{\sigma}$ be an order $p$automorphism of $V_{\sqrt{2}A_{p-1}}$ which is a lift of a $p$-cycle in theWeyl group ${\rm Weyl}(A_{p-1})\cong {\mathfrak S}_p$. We study a certainextension $V$ of a tensor product of finitely many copies of the orbifold model$V_{\sqrt{2}A_{p-1}}^{\langle \widehat{\sigma} \rangle}$ and give a criterionfor $V$ that every irreducible $V$-module is a simple current.

Extensions of tensor products of ${\mathbb Z}_p$-orbifold models of the lattice vertex operator algebra $V_{\sqrt{2}A_{p-1}}$