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In this paper, the complete group classification is performed on the generalized short pulse equation, which includes a lot of important nonlinear wave equations as its special cases. In the sense of geometric symmetry, all of the vector fields of the equation are obtained in terms of the arbitrary functions. Then, the symmetry reductions and exact solutions to the equations are investigated. Especially, we develop the analytic power series method for constructing the exact power series solutions to the short pulse types of equations.