Dimension Formulae in Genus Zero and Uniqueness of Vertex Operator Algebras

We prove a dimension formula for orbifold vertex operator algebras of central charge 24 by automorphisms of order n such that $\Gamma _{0}(n)$ is a genus zero group. We then use this formula together with the inverse orbifold construction for automorphisms of orders 2, 4, 5, 6, and 8 to establish that each of the following fifteen Lie algebras is the weight-one space $V_{1}$ of exactly one holomorphic, $C_{2}$-cofinite vertex operator algebra V of CFT type and central charge 24: $A_{5}C_{5}E_{6,2}$, $A_{3}A_{7,2}{C_{3}^{2}}$, $A_{8,2}F_{4,2}$, $B_{8}E_{8,2}$, ${A_{2}^{2}}A_{5,2}^{2}B_{2}$, $C_{8}{F_{4}^{2}}$, $A_{4,2}^{2}C_{4,2}$, $A_{2,2}^{4}D_{4,4}$, $B_{5}E_{7,2}F_{4}$, $B_{4}{C_{6}^{2}}$, $A_{4,5}^{2}$, $A_{4}A_{9,2}B_{3}$, $B_{6}C_{10}$, $A_{1}C_{5,3}G_{2,2}$, and $A_{1,2}A_{3,4}^{3}$.