  We identify the quantum isometry groups of spectral triples built on the
symmetric groups with length functions arising from the nearest-neighbor
transpositions as generators. It turns out that they are isomorphic to certain
"doubling" of the group algebras of the respective symmetric groups. We discuss
the doubling procedure in the context of regular multiplier Hopf algebras. In
the last section we study the dependence of the isometry group of S_n on the
choice of generators in the case n=3. We show that two different choices of
generators lead to non-isomorphic quantum isometry groups which exhaust the
list of non-commutative non-cocommutative semisimple Hopf algebras of dimension
12. This provides non-commutative geometric interpretation of these Hopf
algebras.
