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