Linear representation theory of groups of order 243

Correspondence between degrees of irreducible representations and conjugacy class sizes
See also linear representation theory of groups of order 243.

For groups of order 243, it is true that the list of conjugacy class sizes completely determines the list of degrees of irreducible representations, though the converse does not hold, i.e., the degrees of irreducible representations need not determine the conjugacy class sizes. The details are given below. The middle column, which is the total number of each, separates the description of the list of conjugacy class sizes and the list of degrees of irreducible representations:

Note that there are two possibilities for the conjugacy class size statistics corresponding to the degrees of irreducible representations with 9 of degree 1 and 26 of degree 3.