The characteristic connection is a good substitute for Levi-Civita connection in studying non-integrable geometries. In this paper we consider the homogeneous space U(3)=(U(1)£U(1)£ U(1)) with a one-parameter family of Hermitian structures. We prove that the one-parameter family of Hermtian structures admit a characteristic connection. We also compute the torsion of the characteristic connecitons.