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$C_2$ cofiniteness and rationality of $V_{L_2}^{S_4}$ are obtained, andirreducible $V_{L_2}^{S_4}$-modules are classified. With the assumption ofrationality and $C_2$ cofiniteness, irreducible $V_{L_2}^{A_5}$-modules aredetermined. Also, quantum dimensions of these irreducible modules arecalculated.

Representations of Vertex Operator Algebras $V_{L_2}^{S_4}$ and $V_{L_2}^{A_5}$