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 $\chi \in \Irr(G)$, 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 $\chi \in \Irr(G)$ that is fully ramified over $Z(G)$. Motivated by this notion, an irreducible character $\chi \in \Irr(G)$ is called of \emph{central type} if $\chi$ vanishes on $G \setminus Z(\chi)$, where \[ Z(\chi)=\{\, g \in G : |\chi(g)|=\chi(1) \,\} \] 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 $\chi,\psi \in \Irr(G)$, 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.