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 $N⊴G$ and $χ\in \Irr(G)$, we say that $\chi$ is \emph{fully ramified} over $N$ if $\chi (g)=0$ 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 $\chi$ vanishes on $G\setminus Z(\chi )$, where \[ Z(χ)=\{\, g \in G : |χ(g)|=χ(1) \,\} \] is the center of $\chi$. 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 $Z(\chi )\subseteq Z(\psi )$ or $Z(\psi )\subseteq Z(\chi )$. 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.