Usage patterns
政権を取るto seize/take power
The opposition party won the election and took power.
政権が倒れるfor a government/administration to fall
The administration collapsed due to a corruption scandal.
政権を維持するto maintain (one's hold on) power
The prime minister tried to maintain his hold on power while pushing reforms.
政権の座に就くto assume power; to take the seat of government
The new president assumed power.
Expressions
政権下under the ... administration; within the ... government 政権を握るcome to power; assume the reins of government