Noether's problem for groups of order 243☆

Let k be any field, G be a finite group. Let G act on the rational function field k(xg:g∈G)k(xg:g∈G) by k  -automorphisms defined by h⋅xg=xhgh⋅xg=xhg for any g,h∈Gg,h∈G. Denote by k(G)=k(xg:g∈G)Gk(G)=k(xg:g∈G)G the fixed field. Noether's problem asks, under what situations, the fixed field k(G)k(G) will be rational (= purely transcendental) over k. According to the data base of GAP there are 10 isoclinism families for groups of order 243. It is known that there are precisely 3 groups G of order 243 (they consist of the isoclinism family Φ10Φ10) such that the unramified Brauer group of C(G)C(G) over CC is non-trivial. Thus C(G)C(G) is not rational over CC. We will prove that, if ζ9∈kζ9∈k, then k(G)k(G) is rational over k for groups of order 243 other than these 3 groups, except possibly for groups belonging to the isoclinism family Φ7Φ7.