The vibronic coupling between the fivefold degenerate orbital representation of the icosahedral point group and vibrational modes which are also fivefold degenerate is characterised by the existence of two equisymmetric but independent channels. It is shown that this coupling multiplicity of the quintuplet Jahn-Teller Hamiltonian can be resolved using the symmetric group of permutations of six elements, S-6, of which the group of rotations of an icosahedron is a subgroup. After the application of this symmetry in atomic physics by Judd and Lo (Phys. Rev. Lett., 82 (1999) 3224), this report forms the first molecular application of this extension of the icosahedral symmetry group. Copyright (c) EPLA, 2007.