In this paper, using the general theory of the Lie groups we concretely discussed the mathematical treatments of the group SU4. The algebra structure of the group SU4, the formula for the dimensions of the irreducible representation, the explicit form of the three elementary representations, and the reduction of direct products of representations arc given. The following symmetry models of the elementary particles based on the group SU4 are built: (1) the model of Baery-Van Hove, (2) the model of Schwinger, (3) the model of Sakata with particle and anti-particle symmetry, (4) the octet model with particle and anti-particle symmetry. In these models some new consequences consistent with the experiments are obtained.