On some metabelian 2-group whose abelianization is of type (2, 2, 2) and applications

Let G   be some metabelian 2-group satisfying the condition G/G′≃ℤ/2ℤ×ℤ/2ℤ×ℤ/2ℤ. In this paper, we construct all the subgroups of G of index 2 or 4, we give the abelianization types of these subgroups and we compute the kernel of the transfer map. Then we apply these results to study the capitulation problem of the 2-ideal classes of some fields k satisfying the condition Gal(k2(2)/k)≃G, where k2(2) is the second Hilbert 2-class field of k.