# Nilpotency class and order determine degrees of irreducible representations for groups up to prime-fourth order

This article gives a fact that is true for small groups of prime power order.More specifically, it is true for all groups of order $p^k$ where $k$ is at most equal to 4.
See more such facts| See more facts true for prime powers up to the same maximum power 4

## Statement

Suppose $p$ is a prime number and $0 \le k \le 4$. Then, for a group of prime power order of order $p^k$, the nilpotency class $c$ of the group determines the degrees of irreducible representations of the group.

Here is the complete list:

$k$ $c$ number of irreducible representations of degree 1 number of irreducible representations of degree $p$
0 0 1 0
1 1 $p$ 0
2 1 $p^2$ 0
3 1 $p^3$ 0
3 2 $p^2$ $p - 1$
4 1 $p^4$ 0
4 2 $p^3$ $p^2 - p$
4 3 $p^2$ $p^2 - 1$