<p>The finite subgroups of are classified up to conjugation in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978); in particular, there exist non-conjugate finite groups in . Each finite group of acts naturally on ; thus we get a faithful -lattice with . In this way, there are exactly such lattices. Given a -lattice with , the group acts on the rational function field by multiplicative actions, i.e. purely monomial automorphisms over . We are concerned with the rationality problem of the fixed field . A tool of our investigation is the unramified Brauer group of the field over . It is known that, if the unramified Brauer group, denoted by , is non-trivial, then the fixed field is not rational (= purely transcendental) over . A formula of the unramified Brauer group for the multiplicative invariant field was found by Saltman in 1990. However, to calculate for a specific multiplicatively invariant field requires additional efforts, even when the lattice is of rank equal to . There is a direct decomposition where is some subgroup of . The first summand , which is related to the faithful linear representations of , has been investigated by many authors. But the second summand doesn’t receive much attention except when the rank is . Theorem 1. Among the finite groups , let be the associated faithful -lattice with , there exist precisely lattices with . In these situations, and thus . The groups are isomorphic to , , , , whose GAP IDs are (4,12,4,12), (4,32,1,2), (4,32,3,2), (4,33,3,1), (4,33,6,1) respectively in Brown, Büllow, Neubüser, Wondratscheck, and Zassenhaus (1978) and in The GAP Group (2008). Theorem 2. There exist (resp. ) finite subgroups in (resp. ). Let be the lattice with rank (resp. ) associated to each group . Among these lattices precisely (resp. ) of them satisfy the condition . The GAP IDs (actually the CARAT IDs) of the corresponding groups may be determined explicitly. Motivated by these results, we construct -lattices of rank , , ( is any positive integer and is any odd prime number) satisfying that and ; and therefore are not rational over . For these -lattices , we prove that the flabby class of is not invertible. We also construct an example of -lattice (resp. -lattice) of rank (resp. ) with . As a consequence, we give a counter-example to Noether’s problem for over where is some abelian group.</p>