Classifying primitive solvable permutation groups of rank \(5\) and \(6\) (Q6634499)

Let \(G\) be a finite solvable permutation group acting faithfully and primitively on a finite set \(\Omega.\) Let \(G_0\) be the stabilizer of a point \(\alpha\) in \(\Omega.\) The rank of \(G\) is defined as the number of orbits of \(G_0\) in \(\Omega,\) including the trivial orbit \(\{\alpha\}.\) The authors completely classify the cases where \(G\) has rank \(5\) and \(6\), continuing the previous works of different authors including \textit{B. Huppert} [Math. Z. 67, 479--518 (1957; Zbl 0079.03701); Math. Z. 68, 126--150 (1957; Zbl 0079.25502)], \textit{D. G. Higman} [Math. Z. 91, 70--86 (1966; Zbl 0136.01402); Math. Z. 104, 147--149 (1968; Zbl 0162.24202)], \textit{C. Hering} [J. Algebra 93, 151--164 (1985; Zbl 0583.20003)], \textit{M. W. Liebeck} [Proc. Lond. Math. Soc. (3) 54, 477--516 (1987; Zbl 0621.20001)], \textit{M. W. Liebeck} and \textit{J. Saxl} [Bull. Lond. Math. Soc. 18, 165--172 (1986; Zbl 0586.20003)], \textit{D. A. Foulser} [Trans. Am. Math. Soc. 143, 1--54 (1969; Zbl 0187.29401)], \textit{M. Dolorfino} et al. [``Classifying solvable primitive permutation groups of low rank'', Preprint, \url{arXiv:2211.16558}], \textit{Y. Yang} [J. Algebra 323, No. 10, 2735--2755 (2010; Zbl 1195.20017); J. Algebra 341, No. 1, 23--34 (2011; Zbl 1247.20021)], \textit{Y. Yang} et al. [J. Algebra 590, 139--154 (2022; Zbl 1523.20024)] on classifying groups of rank \(4\) or lower.