You might want to think deeper about that one. If you tried, you could probably come up with real-life examples of virtually every combination.
"Oligarchy, like communism, is BOTH an economic system AND a form of rule"
I would have to say that statement is false. While communism is usually accompanied by a totalitarian political system (which could be monarchical like some African countries, or democratic like Venezuela), there are examples of communism on a smaller scale, like the Oneida Colonies, where the communist economy is overseen by a republican board of directors. And the textbook ultimate form of communism, is anarchical, with no government at all.
The interactions of the political and economic systems are important, and I think it is best to not oversimplify.
The Oneida Community was in reality ruled by the charismatic personality of its founder, John Noyes, and effectively dissolved after a few decades as a result of the death of its founder. Regardless of the official forms, the reality was that Noyes ran things. And since the community was based in the United States, and subject to the laws thereof, it forms an invalid example.
As far as "textbook ultimate forms", that's where they are found, in textbooks.
In real life, wherever communism is tried as a sovereign political system, it is necessarily totalitarian. It rests on its ability to effectively enslave the productive members of society for the benefit of the rulers of the system, and its ability to prevent those productive members from leaving their slavery.
In countries like Venezuela, you have the situation where the unproductive form a democratic majority, and use their power to strip the productive minority of their wealth. The net result in a few years will be their collapse into poverty. This is why democracy does not work in places where the middle class is a voting minority. This is where the United States looks like its heading.
An oligarchic system uses the political side to enrich the members of the oligarchy and buy their loyalty, and the members of the oligarchy use their wealth and power to maintain the system.