Noether's problem for transitive permutation groups of degree 6

Suppose that a finite group G is realized in the Cremona group Cr-m(k), the group of k-automorphisms of the rational function field K of m variables over a constant field k. The most general version of Noether's problem is then to ask, whether the subfield K-G consisting of G-invariant elements is again rational or not. This paper treats Noether's problem for various subgroups G of G6, the symmetric group of degree 6, acting on the function field Q(s, t, z) over k = Q of the moduli space M-0,(6) of P-1 with ordered six marked points. We shall show that this version of Noether's problem has an affirmative answer for all but two conjugacy classes of transitive subgroups G of G6, by exhibiting explicitly a system of generators of the fixed field Q(s, t, z)G. In the exceptional cases G = 21(6), 21(5), the problem remains open.