Nikolaus Conference 2018

A general question on character degrees is how the set cd(G) of irreducible character degrees of a finite group G reflects and is reflected by the structure of the group. There are several results devoted to studying groups with few character degrees. To aid in the study of the relation between cd(G) and the structure of G, several graphs have been attached to cd(G). The common divisor graph of G, denoted by Γ(G), is a simple graph whose vertices are non-trivial character degrees of G, and two vertices are adjacent if they are not relatively prime. In this talk, we concentrate on non-solvable groups with six character degrees and classify graphs with five vertices that may occur as Γ(G) for some non-solvable group G.