Classification of GVZ and Nested GVZ $p$-groups up to Order $p^6$

Abstract

Let $G$ be a finite group and let $\Irr(G)$ denote the set of irreducible complex characters of $G$. For a normal subgroup $N \trianglelefteq G$ and $χ\in \Irr(G)$, we say that $χ$ is \emph{fully ramified} over $N$ if $χ(g)=0$ for all $g \in G \setminus N$. A group $G$ is said to be of \emph{central type} if there exists $χ\in \Irr(G)$ that is fully ramified over $Z(G)$. Motivated by this notion, an irreducible character $χ\in \Irr(G)$ is called of \emph{central type} if $χ$ vanishes on $G \setminus Z(χ)$, where \[ Z(χ)=\{\, g \in G : |χ(g)|=χ(1) \,\} \] is the center of $χ$. Groups in which every irreducible character is of central type are called \emph{GVZ-groups}. Furthermore, a group $G$ is said to be \emph{nested} if for all $χ,ψ\in \Irr(G)$, either $Z(χ)\subseteq Z(ψ)$ or $Z(ψ)\subseteq Z(χ)$. It is known that a GVZ-group is nilpotent. In this article, we classify all GVZ and nested GVZ $p$-groups of order at most $p^6$, where $p$ is an odd prime.