In this paper, we propose a high-order mode sheet beam extended interaction klystron (EIK) operating at G-band. Through the study of electric field distribution, we choose TM31 2π mode as the operating mode. The eigenmode simulation shows that the resonant frequency of the modes adjacent to the operating mode is far away from the central frequency, so there is almost no mode competition in our high mode EIK. In addition, by studying the sensitivity of the related geometry parameters, we conclude that the height of the coupling cavity has a great influence on the effective characteristic impedance, and the width of the gap mainly affects the working frequency. Therefore, it is necessary to strictly control the fabrication tolerance within 2 μm. Finally, the RF circuit using six barbell multi-gap cavities is determined, with five gaps for the input cavity and idler cavities and seven gaps for the output cavity. To expand the bandwidth, the stagger tuning method is adopted. Under the conditions of a voltage of 16.5 kV, current of 0.5 A and input power of 0.2 W, the peak output power of 650 W and a 3-dB bandwidth of 700 MHz are achieved without any self-oscillation.