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In this article, we give a sufficient and necessary condition for the$C_2$-cofiniteness of the 2-cycle permutation orbifold model $(V\otimesV)^\sigma$ for a $C_2$-cofinite vertex operator algebra and the 2-cyclepermutation $\sigma$ of $V\otimes V$. As an application, we show that the2-cycle permutation orbifold model of the simple Virasoro vertex operatoralgebra $L(c,0)$ of minimal central charge $c$ is $C_2$-cofinite.

C_2-cofiniteness of the 2-cycle permutation orbifold models of minimal Virasoro vertex operator algebras