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Militarism in Japan had been on the rise since the late 19th century, but by the time of WW2, the military was largely independent of civilian oversight and had a large say in who could be appointed to what in the government.
It was a gradual process making it hard to pin down one prime minister since there was no takeover as in a coup, only a slow shift of power to the military's increasing influence. Even with that much power, there were prime ministers that did not follow the military's lead in everything.
But if you're looking for the prime minister appointed by the Emperor when he was convinced by the military that war was inevitable with the West, then you're looking for Hideki Tojo - who was himself a general in the Imperial Japanese Army.