Linear representation theory of groups of prime-fourth order

Full listing
It turns out that nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order, so there are only three possibilities for the degrees of irreducible representations for groups of order 81, based on whether the nilpotency class is 1, 2, or 3.

The case $$p = 2$$ is somewhat different from the others, because there are only three maximal class groups of order $$2^4$$ as opposed to four for odd $$p$$. For the case $$p = 2$$, see linear representation theory of groups of order 16.