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 ${\mathrm{\Gamma }}_0(n)$ 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 of 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 $A_{1,2}{A}_{3,4}^3$.