Linear representation theory of groups of prime-fifth order

Grouping by cumulative sums of squares of degrees
Note that it is true in this case that the sum of squares of degrees of irreducible representations of degree dividing any number itself divides the order of the group (in particular, all these numbers are powers of $$p$$). However, this is not true for all groups and in fact an analogous statement fails for groups of prime-sixth order (see linear representation theory of groups of prime-sixth order). For more, see:


 * All partial sum values of degrees of irreducible representations divide the order of the group for groups up to prime-fifth order
 * There exist groups of prime-sixth order in which the partial sum values of degrees of irreducible representations do not divide the order of the group

Correspondence between degrees of irreducible representations and conjugacy class sizes
See also element structure of groups of prime-fifth order.

For groups of order $$p^5$$, it is true that the list of conjugacy class sizes determines the degrees of irreducible representations. In the case $$p = 2$$, the converse also holds, i.e., the degrees of irreducible representations determine the conjugacy class sizes.

However, for $$p \ge 3$$, there is one ambiguous case: the case of $$p^2$$ degree one and $$p^3 - 1$$ degree two representations corresponds to two possible lists of conjugacy class sizes: ($$p$$ of size one, $$p^3 - 1$$ of size $$p$$, $$p^2 - p$$ of size $$p^3$$), and ($$p^2$$ of size 1, $$p^3 - 1$$ of size $$p^2$$). For $$p = 2$$, there are no groups fitting the latter case.