BICYCLIC COMMUTATOR QUOTIENTS WITH ONE NON-ELEMENTARY COMPONENT

For any number field K with non-elementary 3-class group Cl-3(K).(sic) C-3(e) xC(3), e greater than or less than 2, the punctured capitulation type n(K) of K in its unramified cyclic cubic extensions Li, 1i 1 less than or greater than 1 less than or greater than 4, is an orbit under the action of S3 xS3. By means of Artin's reciprocity law, the arithmetical invariant n(K) is translated to the punctured transfer kernel type {(G2) of the automorphism group G2 = Gal(F2 3 (K)/K) of the second Hilbert 3-class field of K. A classification of finite 3-groups G with low order and bicyclic commutator quotient G/G'. C3e x C3, 2 6 e 6 6, according to the algebraic invariant n(G), admits conclusions concerning the length of the Hilbert 3-class field tower F-infinity (3) (K) of imaginary quadratic number fields K.