  For a prime \(p\ge 2\) and a number field K with p-class group of type (p,p)
it is shown that the class, coclass, and further invariants of the metabelian
Galois group \(G=Gal(F_p^2(K) | K)\) of the second Hilbert p-class field
\(F_p^2(K)\) of K are determined by the p-class numbers of the unramified
cyclic extensions \(N_i | K\), \(1\le i\le p+1\), of relative degree p. In the
case of a quadratic field \(K=\mathbb{Q}(\sqrt{D})\) and an odd prime \(p\ge
3\), the invariants of G are derived from the p-class numbers of the non-Galois
subfields \(L_i | \mathbb{Q}\) of absolute degree p of the dihedral fields
\(N_i\). As an application, the structure of the automorphism group
\(G=Gal(F_3^2(K) | K)\) of the second Hilbert 3-class field \(F_3^2(K)\) is
analysed for all quadratic fields K with discriminant \(-10^6<D<10^7\) and
3-class group of type (3,3) by computing their principalisation types. The
distribution of these metabelian 3-groups G on the coclass graphs G(3,r),
\(1\le r\le 6\), in the sense of Eick and Leedham-Green is investigated.
