On the second 2-class group Gal(K2(2)/K) of some imaginary quartic cyclic number field K

Let H8 denote the quaternion group of order 8 and put G=Z/2Z×H8. Let K be some imaginary quartic cyclic number field whose 2-class group is of type (2,2,2). In this paper, we prove in particular that G is realizable over K, i.e., G≃Gal(K2(2)/K) where K2(2) is the second Hilbert 2-class field of K. Then we study the capitulation problem of the 2-ideal classes of K in the fourteen intermediate unramified extensions between K and its first Hilbert 2-class field. Additionally, these fourteen unramified extensions are constructed, and the abelian type invariants of their 2-class groups and the length of the 2-class field tower of K are given.