C₂-Cofiniteness of Permutation Orbifolds I: Relative C₁ Descent

We establish a relative C₁-cofiniteness criterion for C₂-cofiniteness of finite group orbifolds. For permutation actions on a simple, self contragredient, C₂-cofinite vertex operator algebra of CFT type, it suffices to check a single relation for a derivative of the conformal vector in the Virasoro vertex operator algebra. This reduces the problem to a calculation depending only on the central charge and tensor degree. An exact calculation then proves cofiniteness for a family of nonsolvable permutation orbifolds without restricting the central charge.