  For a transitive subgroup $G \le S_6$ which contain $C_3 \times C_3$ as
subgroup, we prove that $K(x_1,\dots,x_6)^G$ is rational over $K$, where $K$ is
any field, and $G$ acts naturally on $K(x_1,\dots,x_6)$ by permutations on the
variables. We also give an application on construction of generic polynomials.
