Invariants of wreath products and subgroups of S6

Let G be a subgroup of S-6, the symmetric group of degree 6. For any field k, G acts naturally on the rational function field k(x(1),...,x(6)) via k-automorphisms defined by sigma center dot x(i) = x(sigma(i)) for any sigma is an element of G and any 1 <= i <= 6. We prove the following theorem. The fixed field k(x(1),..., x(6))(G) is rational (i.e., purely transcendental) over k, except possibly when G is isomorphic to PSL2 (F-5), PGL(2) (F-5), or A(6). When G is isomorphic to PSL2 (F-5) or PGL(2) (F-5), then C(x(1),...,x(6))(G) is C-rational and k(x(1),...,x(6))(G) is stably k-rational for any field k. The invariant theory of wreath products will be investigated also.