On some realizable metabelian 5-groups

Abstract

Let G be a 5-group of maximal class and G' = [G, G] its derived group. Assume that the abelianization G/G' is of type (5, 5) and the transfers from H1 to G' and from H2 to G' are trivial, where H1 and H2 are two maximal normal subgroups of G. Then G is completely determined with the isomorphism class groups of maximal class. Moreover the group G is realizable with some fields k, which is the normal closure of a pure quintic field.