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A new conceive of high-order mode multi-beam Extended interaction Klystron (EIK) at the terahertz (THz) band is proposed. The interaction circuit based on the coaxial coupled cavity at <inline-formula> <tex-math notation="LaTeX">${\mathrm {TM}}_{\mathrm {10,1,0}} - 2\pi $ </tex-math></inline-formula> mode is designed and analyzed. Benefiting from the larger transverse space of the coaxial coupled cavity operating at high-order mode, multiple electron beams are formatted to reduce the current density of each electron beam. The crucial parameters associated with the interaction are determined by theoretical calculation and optimized by three-dimensional (3D) Particle-in-Cell (PIC) simulation. The results of theoretical analysis and PIC simulation demonstrate that the EIK operates at <inline-formula> <tex-math notation="LaTeX">${\mathrm {TM}}_{\mathrm {10,1,0}} - 2\pi $ </tex-math></inline-formula> mode without mode competition and self-oscillation. Driven by ten individual electron beams with the voltage of 20 kV, the current of 0.1 A and the current density of 318 A/cm², the maximum gain of 48 dB is achieved at the center frequency of 300 GHz with a 3-dB gain bandwidth of 200 MHz. And the corresponding output power and efficiency are 312.5 W and 1.56 &#x0025;. The results indicate that the proposed EIK is a practical approach to generate high power terahertz waves using the thermal beam with a low compression ratio.