I wouldn't use a dictionary definition here, just as little as I would use a dictionary definition to explain the Theory of Relativity. Especially since fascism is much more something akin to a meta-ideology than a system of government.
As for racism and fascism, racism is not necessarily a part of it. But fascism is dependent on inner and outer enemies that must be declared, blamed, cleansed, and victimised. And racism is one of the most readily usable tools for that.