Designing a good preconditioner for accelerating the iterative solution of the three-dimensional multi-group radiation diffusion equations based on a cell-centered finite volume discretization has been the focus of intensive research efforts over the past few decades. In the present paper, we develop a physics-wise splitting preconditioning algorithm with selective relaxation and algebraic multigrid subsolves. The spectral distribution and the degree of the minimal polynomial of its right-preconditioned matrix together with the conditional convergence property of its iteration method are analyzed. Subsequently, we discuss its sequential implementation as well as the two-level parallelization. Lastly, the new preconditioner is applied to the experimental test cases arising from realistic simulations of the hydrodynamic instability during the deceleration phase of a laser-driven spherical implosion to illustrate the numerical robustness, computational efficiency, parallel strong and weak scalabilities, and its competitiveness with some existing monolithic and block preconditioning approaches.