An improved adjoint operator method is employed to compute the third order normal form of six dimensional nonlinear dynamical systems and the associated nonlinear transformation for the first time. First an improved adjoint operator method is briefly introduced. Then, a general six dimensional nonlinear system is analyzed to drive the formula of computing the third order normal form. Finally, the MAPLE symbolic program for calculating the third order normal form is given. The concrete normal form of six dimensional nonlinear systems is obtained. The results obtained here indicate that we may respectively obtain the normal forms, the coefficients of the normal forms and the associated near identity nonlinear transformations for six dimensional nonlinear systems in four different cases by using a same main Maple symbolic program.