Vertex operator algebras generated by two conformal vectors whose τ-involutions generate S3

We study a vertex operator algebra (VOA) V of moonshine type with two τ-involutions φ and ψ which generate S3. In this case, V contains rational conformal vectors e and f with central charge 1/2 such that φ=τe, ψ=τf, and eτfτe=f. We determined the inner products e,f for such conformal vectors e and f and showed a subVA generated by e and f is a VOA with central charge 1/2+20/21 or 4/5+6/7 and has a Griess algebra of dimension three or four, respectively.