We present the Drazin-inverse solution of the matrix equation $AXB=G$ as a least-squares solution of a specified minimization problem. Some important properties of the Moore--Penrose inverse are extended on the Drazin inverse by exploring the minimal norm properties of the Drazin-inverse solution of the matrix equation $AXB=G$. The least squares properties of the Drazin-inverse solution lead to new representations of the Drazin inverse of a given matrix, which are justified by illustrative examples.