This monograph addresses the problem of describing allprimitive soluble permutation groups of a given degree, withparticular reference to those degrees less than 256. Thetheory is presented in detail and in a new way using modernterminology. A description is obtained for the primitivesoluble permutation groups of prime-squared degree and apartial description obtained for prime-cubed degree. Thesedescriptions are easily converted to algorithms forenumerating appropriate representatives of the groups. Thedescriptions for degrees 34 (die vier hochgestellt,Sonderzeichen) and 26 (die sechs hochgestellt,Sonderzeichen) are obtained partly by theory and partly bymachine, using the software system Cayley.The material is appropriate for people interested in solublegroups who also have some familiarity with the basictechniques of representation theory.This work complements the substantial work already done oninsoluble primitive permutation groups. Background theory.- The imprimitive soluble subgroups of GL(2, p k ).- The normaliser of a Singer cycle of prime degree.- The irreducible soluble subgroups of GL(2, p k ).- Some irreducible soluble subgroups of GL(q, p k ), q>2.- The imprimitive soluble subgroups of GL(4, 2) and GL(4, 3).- The primitive soluble subgroups of GL(4, p k).- The irreducible soluble subgroups of GL(6, 2).- Conclusion.- The primitive soluble permutation groups of degree less than 256.
