Using the first principle full-potential linearized augmented wave method we study the electronic structure and elastic and magnetic properties of CoMnZn Z( Z= Si, Ge, Sn, Pb) LiMgPdSn-type Heusler alloys. These compounds have the composition CoMnZn Zwith 1︰1︰1︰1 stoichiometry, where Zdenotes the main group element Si, or Ge, or Sn, or Pb. The exchange-correlations are treated within the generalized gradient approximation of Perdewe-Burke-Ernzerhof. For each of all studied Heusler alloys, the ferromagnetic state is considered to be more stable than the paramagnetic state, judged by the energy. The total energy of the magnetic calculation is lower than that of the nonmagnetic state for each of all three serise compounds at the equilibrium lattice constant, indicating that the magnetic state is more stable than the nonmagnetic state. We determine the elastic constants C ₁₁, C ₁₂and C ₄₄, which have not been established previously in experiment nor in theory. The elastic constant indicates the weakened resistance to sheardeformation compared with the resistance to unidirectional compression. We derive other mechanical parameters, i.e., the shear modulus G, Young’s modulus E, Poisson’s ratio ν, and shear anisotropic factor A, which are the important elastic moduli for applications. These compounds each have a lower anisotropy and possess a low probability to develop micro-crack or structural defect in its growing process. The sound velocity and Debye temperature for each of the CoMnZn Z( Z= Si, Ge, Sn, Pb) compounds in their stable structure are calculated. The CoMnZnPb exhibits the lowest Debye temperature, and the highest value is observed for CoMnZnGe. The electronic structure calculations show that CoMnZn Z( Z= Si, Ge, Sn) each exhibit a gap in the band of minority states, and they are clearly half-metallic ferromagnets, except for the CoMnZnPb. The CoMnZn Z( Z= Si, Ge, Sn) compounds and their magnetic moments are in reasonable agreement with the Slater-Pauling rule, and they comply with a Slater-Pauling rule of M _t= Z _t– 28, which indicates the half metallicity and high spin polarization for these compounds. The CoMnZnSi compound has the largest half-metallic gap value and the gap is about 0.66 eV. The magnetic properties are primarily determined by the Mn atoms, which contribute the highest magnetic moments. The localmoment of the Zelement atom is negligibly small. The hybridization of the d orbitals between Co and Mn can explain the origin of the Slater-Pauling rule in half-metallic quaternary Heusler alloys. The half-metallic gap comes mainly from the interaction between the Co and Mn atoms.