AbstractLet G be a p-group, and let χ be an irreducible character of G.
The codegree of χ is given by {\lvert G:\operatorname{ker}(\chi)\rvert/\chi(1)}.
Du and Lewis have shown that a p-group with exactly three codegrees has nilpotence class at most 2.
Here we investigate p-groups with exactly four codegrees.
If, in addition to having exactly four codegrees, G has two irreducible character degrees, G has largest irreducible character degree {p^{2}}, {\lvert G:G^{\prime}\rvert=p^{2}}, or G has coclass at most 3, then G has nilpotence class at most 4.
In the case of coclass at most 3, the order of G is bounded by {p^{7}}.
With an additional hypothesis, we can extend this result to p-groups with four codegrees and coclass at most 6.
In this case, the order of G is bounded by {p^{10}}.
Character degrees in blocks and defect groups
