  On one hand the Geroch group allows one to associate spacetime independent
matrices with gravitational configurations that effectively only depend on two
coordinates. This class includes stationary axisymmetric four- and
five-dimensional black holes. On the other hand, a recently developed inverse
scattering method allows one to factorize these matrices to explicitly
construct the corresponding spacetime configurations. In this work we
demonstrate the construction as well as the factorization of Geroch group
matrices for a wide class of black hole examples. In particular, we obtain the
Geroch group SL(3,R) matrices for the five-dimensional Myers-Perry and
Kaluza-Klein black holes and the Geroch group SU(2,1) matrix for the
four-dimensional Kerr-Newman black hole. We also present certain non-trivial
relations between the Geroch group matrices and charge matrices for these black
holes.
