Under this system the government owns nearly all the means of production and directs all important economic activities?

The form of government is called: communism.

The "people" are nominally the joint owners of all goods and products within the system. However, because the system is operated by a government apparatus (to ensure complete equality), the outcome has invariably been complete ownership by the state instead, and by its leadership. This also includes controlling most aspects of the lives of the populace. What appeared to Karl Marx to be an ideal and utopian existence has the fatal human "flaws" of self-interest, self-determination, and independent thought.