Noether's problem for some subgroups of S14: The modular case

Let $G$ be a subgroup of $S_{n}$, the symmetric group degree $n$. For any field $k$, acts naturally on rational function $k(x_{1},\cdots,x_{n})$ via $k$-automorphisms defined by $\sigma\cdot x_{i}:=x_{\sigma\cdot i}$ for $\sigma\in G$ and $1\leq i\leq n$. In this article, we will show that if is solvable transitive $S_{14}$ $\text{char}(k)=7$, then fixed subfield $k(x_{1},\cdots,x_{14})^{G}$ (i.e., purely transcendental) over $k$. proving above theorem, rely Kuniyoshi-Gasch\"utz Theorem or some ideas in its proof.