Let $p$ be an odd prime. We construct a $p$-group $P$ of nilpotency class two, rank seven and exponent $p$, such that $\text{Aut}(P)$ induces $N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$ on the Frattini quotient $P/\unicode[STIX]{x1D6F7}(P)$. The constructed group $P$ is the smallest $p$-group with these properties, having order $p^{14}$, and when $p=3$ our construction gives two nonisomorphic $p$-groups. To show that $P$ satisfies the specified properties, we study the action of $G_{2}(q)$ on the octonion algebra over $\mathbb{F}_{q}$, for each power $q$ of $p$, and explore the reducibility of the exterior square of each irreducible seven-dimensional $\mathbb{F}_{q}[G_{2}(q)]$-module.

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与 Chevalley 集团相关的具有自形态群的 ON 群

让$p$是一个奇数素数。我们构建一个$p$-团体$P$幂等二、七阶和指数$p$, 这样$\text{Aut}(P)$诱导$N_{\text{GL}(7,p)}(G_{2}(p))=Z(\text{GL}(7,p))G_{2}(p)$关于弗拉蒂尼商$P/\unicode[STIX]{x1D6F7}(P)$. 构造组$P$是最小的$p$- 具有这些属性的组，具有顺序$p^{14}$， 什么时候$p=3$我们的构造给出了两个非同构的$p$-团体。为了表明$P$满足指定的性质，我们研究的作用$G_{2}(q)$关于八元数代数$\mathbb{F}_{q}$, 对于每个幂$q$的$p$，并探索每个不可约七维的外平方的可约性$\mathbb{F}_{q}[G_{2}(q)]$-模块。