A compact relativistic magnetron with axial output for TE₁₀ mode was investigated. This magnetron with 6 resonant cavities operates at πmode and can well transit between resonant cavities and rectangular output waveguide through the structure design of 4 cavities axial output, which realizes the microwave output of rectangular TE₁₀ mode. Compared with the magnetron with diffraction output (MDO), this design not only makes both the diameter and the axial length of the magnetron minimized so that the whole system can be more compact, but also provides a much purer output mode of a low order in the output waveguide. The operating performance of the magnetron has been analyzed and optimized by using particle-in-cell simulation software. Simulation results show that the power conversion efficiency can reach to 25.0%, corresponding to the output power of 10GW and the working frequency of 2.58GHz, under the condition of the applied voltage of 500kV and the applied magnetic field of 0.5T.