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Let $G=p^{1+2n}{.}Q$, $n>1$, be a finite extension group of an extra-special $p$-group $N=p^{1+2n}$ by a group $Q$, where $Q$ is a linear group of dimension $2n$ over $GF(p)$. It seems that the action (by conjugation) of $G$ (or $Q$) on the $p-1$ irreducible faithful characters of $N$ is transitive. If it is the case, how does one show it?

Let $G=p^{1+2n}{.}Q$, $n>1$, be a finite extension group of an extra-special $p$-group $N=p^{1+2n}$ by a group $Q$, where $Q$ is a linear group of dimension $2n$ over $GF(p)$. It seems that the action (by conjugation) of $G$ (or $Q$) on the $p-1$ irreducible faithful characters of $N$ is transitive. If it is the case, how does one show it? For example, let we consider the maximal subgroup $3_+^{1+4}{:}4S_6$ of $Co_3$.

Let $G=p^{1+2n}{.}Q$ be a finite extension group of an extra-special $p$-group $N=p^{1+2n}$ by a group $Q$, where $Q$ is a linear group of dimension $2n$ over $GF(p)$. It seems that the action (by conjugation) of $G$ (or $Q$) on the $p-1$ irreducible faithful characters of $N$ is transitive. If it is the case, how does one show it?

Let $G=p^{1+2n}{.}Q$, $n>1$, be a finite extension group of an extra-special $p$-group $N=p^{1+2n}$ by a group $Q$, where $Q$ is a linear group of dimension $2n$ over $GF(p)$. It seems that the action (by conjugation) of $G$ (or $Q$) on the $p-1$ irreducible faithful characters of $N$ is transitive. If it is the case, how does one show it?

Let $G=p^{1+2n}{.}Q$ be a finite extension group of an extra-special $p$-group $N=p^{1+2n}$ by a group $Q$. It seems that the action (by conjugation) of $G$ (or $Q$) on the $p-1$ irreducible faithful characters of $N$ is transitive. If it is the case, how does one show it?

Let $G=p^{1+2n}{.}Q$ be a finite extension group of an extra-special $p$-group $N=p^{1+2n}$ by a group $Q$, where $Q$ is a linear group of dimension $2n$ over $GF(p)$. It seems that the action (by conjugation) of $G$ (or $Q$) on the $p-1$ irreducible faithful characters of $N$ is transitive. If it is the case, how does one show it?