Classifying primitive solvable permutation groups of rank 5 and 6

Let 𝐺 be a finite solvable permutation group acting faithfully and primitively on a finite set Ω. Let <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> G₀ be the stabilizer of a point 𝛼 in Ω. The rank of 𝐺 is defined as the number of orbits of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:msub> <m:mi>G</m:mi> <m:mn>0</m:mn> </m:msub> </m:math> G₀ in Ω, including the trivial orbit <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mrow> <m:mo stretchy="false">{</m:mo> <m:mi>α</m:mi> <m:mo stretchy="false">}</m:mo> </m:mrow> </m:math> \{α\} . In this paper, we completely classify the cases where 𝐺 has rank 5 and 6, continuing the previous works on classifying groups of rank 4 or lower.