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.

This monograph addresses the problem of describing all primitive soluble permutation groups of a given degree, with particular reference to those degrees less than 256. The theory is presented in detail and in a new way using modern terminology. A description is obtained for the primitive soluble permutation groups of prime-squared degree and a partial description obtained for prime-cubed degree. These descriptions are easily converted to algorithms for enumerating appropriate representatives of the groups. The descriptions for degrees 34 (die vier hochgestellt, Sonderzeichen) and 26 (die sechs hochgestellt, Sonderzeichen) are obtained partly by theory and partly by machine, using the software system Cayley. The material is appropriate for people interested in soluble groups who also have some familiarity with the basic techniques of representation theory. This work complements the substantial work already done on insoluble primitive permutation groups.