Theorem a. (for the proof see [1])
A real quadratic field K with 3class group of type (3,3)
and capitulation type a.2 (1,0,0,0) or a.3 (0,1,0,0) determines
the metabelian 3group G = Gal(K_{2}K) of automorphisms of the second Hilbert 3class field K_{2} of K
and the symbolic order M of the main commutator [y,x] of G = <x,y> up to a single infinite degree of freedom as
G = G_{0}^{m}(1,0) in CF^{2a}(m), for capitulation type a.2,
G = G_{0}^{m}(0,+1) in CF^{2a}(m), for capitulation type a.3,
with even m ≥ 4, and M = Y_{m2}.

The following table visualizes the begin of this series with a single infinite degree of freedom,
listing the index m of nilpotency, the symbolic order M, the group G, the commutator subgroup G´,
and the 3class numbers of the four associated nonGalois cubic fields.
State  m  M  G  G´  (h(1),...,h(4))  Minimal occurrence 
Ground state  4  Y_{2}  CF^{2a}(4)  (3,3)  (3,3,3,3)  32009 
1^{st} excited state  6  Y_{4}  CF^{2a}(6)  (9,9)  (9,3,3,3)  494236 
2^{nd} excited state  8  Y_{6}  CF^{2a}(8)  (27,27)  (27,3,3,3)  unknown 
...  ...  ...  ...  ...  ...  ... 

*

Theorem c. (for the proof see [1])
A real quadratic field K with 3class group of type (3,3)
and capitulation type c.18 (2,0,1,1) or c.21 (1,2,0,3) determines
the metabelian 3group G = Gal(K_{2}K) of automorphisms of the second Hilbert 3class field K_{2} of K
and the symbolic order M of the main commutator [y,x] of G = <x,y> up to a single infinite degree of freedom as
G = G_{0}^{m,m+1}(0,1,0,1) in CBF^{2a}(m,m+1), for capitulation type c.18,
G = G_{0}^{m,m+1}(0,0,0,1) in CBF^{2a}(m,m+1), for capitulation type c.21,
with odd m ≥ 5, and M = X_{m2}.

The following table visualizes the begin of this series with a single infinite degree of freedom,
listing the index m of nilpotency, the symbolic order M, the group G, the commutator subgroup G´,
and the 3class numbers of the four associated nonGalois cubic fields.
State  m  M  G  G´  (h(1),...,h(4))  Minimal occurrence 
Ground state  5  X_{3}  CBF^{2a}(5,6)  (9,3,3)  (9,3,3,3)  540365 
1^{st} excited state  7  X_{5}  CBF^{2a}(7,8)  (27,9,3)  (27,3,3,3)  1001957 
2^{nd} excited state  9  X_{7}  CBF^{2a}(9,10)  (81,27,3)  (81,3,3,3)  unknown 
...  ...  ...  ...  ...  ...  ... 

*

Theorem E. (for the proof see [1])
A complex or real quadratic field K with 3class group of type (3,3)
and capitulation type E.6 (1,1,2,2) or E.8 (1,2,3,1)
or E.9 (1,2,1,3) or E.14 (2,3,1,1) determines
the metabelian 3group G = Gal(K_{2}K) of automorphisms of the second Hilbert 3class field K_{2} of K
and the symbolic order M of the main commutator [y,x] of G = <x,y> up to a single infinite degree of freedom as
G = G_{0}^{m,m+1}(1,1,1,1) in CBF^{2a}(m,m+1), for capitulation type E.6,
G = G_{0}^{m,m+1}(1,0,1,1) in CBF^{2a}(m,m+1), for capitulation type E.8,
G = G_{0}^{m,m+1}(0,0,+1,1) in CBF^{2a}(m,m+1), for capitulation type E.9,
G = G_{0}^{m,m+1}(0,1,+1,1) in CBF^{2a}(m,m+1), for capitulation type E.14,
with even m ≥ 6, and M = X_{m2}.

The following table visualizes the begin of this series with a single infinite degree of freedom,
listing the index m of nilpotency, the symbolic order M, the group G, the commutator subgroup G´,
and the 3class numbers of the four associated nonGalois cubic fields.
State  m  M  G  G´  (h(1),...,h(4))  Complex example  Real example 
Ground state  6  X_{4}  CBF^{2a}(6,7)  (9,9,3)  (9,3,3,3)  9748  342664 
1^{st} excited state  8  X_{6}  CBF^{2a}(8,9)  (27,27,3)  (27,3,3,3)  262744  unknown 
2^{nd} excited state  10  X_{8}  CBF^{2a}(10,11)  (81,81,3)  (81,3,3,3)  unknown  unknown 
...  ...  ...  ...  ...  ...  ...  ... 

