There are Hopf algebras whose representations exhibit some curious behavior in comparison to those of finite groups and finite group schemes: A tensor power of a nonprojective module may be projective, and the tensor product of a pair of modules may be projective in one order while nonprojective in the other. In this talk we will give a very simple construction of such Hopf algebras from finite groups, explain these phenomena and more, and classify thick tensor left, right, and two-sided ideals in the stable module category. These are all consequences of a theory of support varieties for modules that holds in this setting. This is joint work with Dave Benson.