The characteristic connection is a good substitute for Levi-Civita connection in studying non-integrable geometries. In this paper we consider the homogeneous space $U(3)/(U(1){ imes}U(1){ imes}U(1))$ 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.