  Let p be a prime. For any finite p-group G, the deep transfers T(H,G'):H/H'
--> G'/G'' from the maximal subgroups H of index (G:H)=p in G to the derived
subgroup G' are introduced as an innovative tool for identifying G uniquely by
means of the family of kernels kappa_d(G)=(ker(T(H,G')))_{(G:H)=p}. For all
finite 3-groups G of coclass cc(G)=1, the family kappa_d(G) is determined
explicitly. The results are applied to the Galois groups G=Gal(F_3^\infty/F) of
the Hilbert 3-class towers of all real quadratic fields F=Q(d^1/2) with
fundamental discriminants d>1, 3-class group Cl_3(F)~C_3*C_3, and total
3-principalization in each of their four unramified cyclic cubic extensions
E/F. A systematic statistical evaluation is given for the complete range
1<d<5*10^6, and a few exceptional cases are pointed out for 1<d<64*10^6.
