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Explanation: c'est une prise de pouvoir militaire.
or nous avons affaire à une prise de pouvoir militaire.
Notice if you say prise du pouvoir militaire, it then means that the military is taken over as Alekos mentioned it. Whereas prise de pouvoir militaire means that the military is taking over. The difference is made by the use of du or de. In your case it seems that the military is taking over, hence prise de pouvoir militaire or prise de pouvoir par l'armée