Finitely generated and solvable not implies polycyclic

Statement
It is possible to have a finitely generated solvable group that is not a polycyclic group, and hence, not a Noetherian group, i.e., it has a subgroup that is not finitely generated.

Related facts

 * Finitely generated not implies Noetherian

A general construction using a restricted wreath product
Let $$A$$ be a nontrivial finitely generated solvable group. Let $$G$$ be the uses as intermediate construct::restricted external wreath product of $$A$$ and the uses as intermediate construct::group of integers $$\mathbb{Z}$$ acting regularly. In other words, $$G$$ is the uses as intermediate construct::external semidirect product of $$H$$ and $$\mathbb{Z}$$, where $$H$$ is the uses as intermediate construct::restricted external direct product of countably many copies of $$A$$ and $$\mathbb{Z}$$ acts on the coordinates by a shift of one.

Some examples based on the general construction and otherwise

 * The wreath product of group of integers with group of integers is a finitely generated solvable group that is not polycyclic. The base of the semidirect product here is a countable restricted direct power of the group of integers, which is not finitely generated.
 * Baumslag-Solitar group:BS(1,2) is a finitely generated solvable group that is not polycyclic. The derived subgroup of this group is isomorphic to the additive group of 2-adic rational numbers, which is not finitely generated.