Let k be any field, G be a finite group. Let G act on the rational function field k(x(g) : g is an element of G) by k-automorphisms defined by h . x(g) = x(hg) for any g, h is an element of G. Denote by k(G) = k(x(g) : g is an element of G)(G), the fixed subfield. Noether's problem asks whether k(G) is rational (= purely transcendental) over k. The unramified Brauer group Br-nr(C(G)) and the unramified cohomology H-nr(3)(C(G), Q/Z) are obstructions to the rationality of C(G) (see [14] and [5]). Peyre proves that, if p is an odd prime number, then there is a group G such that vertical bar G vertical bar = p(12), Br-nr(C(G)) = {0}, but H-nr(3)(C(G), Q/Z) not equal {0}; thus C(G) is not stably C-rational [12]. Using Peyre's method, we are able to find groups G with vertical bar G vertical bar = p(9) where p is an odd prime number such that Br-nr(C(G)) = {0}, H-nr(3) (C(G), Q/Z) not equal {0}. (C) 2016 Elsevier Inc. All rights reserved.