The rank 3 permutation representations of finite groups of type \(G_ 2\)
A group \(G^*\) is said to be of type \(G_ 2\) provided \(G_ 2(q)\leq G^*\leq Aut(G_ 2(q)),\) where \(q=p^ f\), p is a prime. In this paper the author proves the following Main Theorem: There exists no faithful primitive permutation representation of rank 3 of \(G^*\) with \(p\geq 3\), except when \(G^*=G_ 2(3)\) and the degrees of such permutation representations are equal to 351. The last section of the paper contains two examples which are inequivalent primitive permutation representations of rank 3 of \(G_ 2(3)\). The proof of the main theorem depends on some properties of complex characters of finite groups of Lie type \(G_ 2(q)\).
faithful primitive permutation representation of rank 3
scientific article; zbMATH DE number 3847640