Let [Formula: see text] denote the discriminant of a real quadratic field. For all bicyclic biquadratic fields [Formula: see text], having a [Formula: see text]-class group of type [Formula: see text], the possibilities for the isomorphism type of the Galois group [Formula: see text] of the second Hilbert [Formula: see text]-class field [Formula: see text] of [Formula: see text] are determined. For each coclass graph [Formula: see text], [Formula: see text], in the sense of Eick, Leedham-Green, Newman and O’Brien, the roots [Formula: see text] of even branches of exactly one coclass tree and, in the case of even coclass [Formula: see text], additionally their siblings of depth [Formula: see text] and defect [Formula: see text], turn out to be admissible. The principalization type [Formula: see text] of [Formula: see text]-classes of [Formula: see text] in its four unramified cyclic cubic extensions [Formula: see text] is given by [Formula: see text] for [Formula: see text], and by [Formula: see text] for [Formula: see text]. The theory is underpinned by an extensive numerical verification for all [Formula: see text] fields [Formula: see text] with values of [Formula: see text] in the range [Formula: see text], which supports the assumption that all admissible vertices [Formula: see text] will actually be realized as Galois groups [Formula: see text] for certain fields [Formula: see text], asymptotically.

Let G be a 3-class group of maximal class, and γ2(G) = [G,G] its derived group. Assume that the commutator factor group G ∕ γ2(G) is of type (3,3) and the transfers Vχ2(G) → γ2(G) and VH→ γ2(G) are trivial, where χ2(G) is the biggest subgroup of G such that [χ2(G), γ2(G)] ⊆ γ4(G), and, H is one of its maximal normals subgroups different to χ2(G). Then G is completely determined with the isomorphism class groups of maximal class defined by B.Nebelung in [24]. Moreover the group G is realised. At the end numerical examples illustrating the results are given.