An S-band compact A6 relativistic magnetron which operates in the π-mode is proposed, and its radiation can be extracted as a TE10 mode axially through a rectangular waveguide which is designed to match radially with the anode. Compared with the magnetron with diffraction output (MDO), the magnetron of this structure can reduce the dimension of the output port of the magnetron, and therefore reduce the volume of the coil of the applied magnetic field, and also reduce the distance between the electron dump and the anode block so that both the diameter and the axial length of the magnetron can be minimized. In addition, as a result of the limit to the dimension of the output waveguide, such a simple converter of the magnetron can provide a much purer radiated mode. The softwares of high-frequency field analysis and CHIPIC particle-in-cell (PIC) simulation have been used to analyze and optimize the operating performance of the magnetron proposed in this paper. Simulation results show that under the condition of the applied voltage of 360 kV, the input power of 1.87 GW and the applied magnetic field of 0.4 T, the output microwave power is 409.2 MW and the power conversion efficiency can reach 21.9%, while the resonant frequency of π-mode is 2.5 GHz and the radiated mode is TE10 in the rectangular waveguide.