Extended interaction klystron is a very important millimeter-wave and sub-millimeter-wave vacuum electron device with many actual and potential applications, such as space-borne cloud profiling radar, communication, imaging system, precision guided missiles, etc. Kinematical theory and space charge wave theory are extensively used to analyze the bunching process of electrons. Kinematical theory is precise when electron beam is especially small because the influence of space charge effect is ignored, while space charge wave theory is accurate when the modulation of electron beam is small since it is based on the premise of small amplitude. Based on kinematical theory, law of induce current, principle of charge conservation in a one-dimensioanl mode and small signal condition, the influence of electron beam on standing wave electric field in multiple-gap cavity is analyzed, and the expression of beam-loading conductance and beam-loading susceptance in multiple-gap cavity are derived. The influence of the direct current transmit angle of single gap, the number of multiple gaps and the direct current transmit angle of between center of adjacent gaps on beam-loading conductance and beam-loading susceptance are analyzed. The results show that the beam-loading conductance and beam-loading susceptance of multiple-gap cavity can change to a bigger extent when the number of cavity gaps is bigger, which means that the maximum beam-wave conversion efficiency and the range of loaded frequency increase with the number of cavity gaps increasing. The results also show that the direct current transmit angle between centers of adjacent gaps is the most important parameter for the beam-wave interaction effect. Based on the above analysis, a G-band extended interaction klystron amplifier consisting of three five-gap cavities is designed by an three-dimensional PIC code. An output power of 225.5 W at 217.94 GHz with an efficiency of 6.26%, whose gain and 3 dB bandwidth are 30.5 dB and 470 MHz respectively, is obtained by simulation. This study is of great significance for the physical design and process in engineering the G-band extended interaction klystron amplifier.