[GAP Forum] Alternative Methods for Constructing Non-Virtually Nilpotent Polycyclic Groups in GAP

[Jonathan Gryak]

Hello Forum Members,
The GAP package Polycyclic contains the function MaximalOrderByUnitsPcpGroup
<http://www.gap-system.org/Manuals/pkg/polycyclic-2.11/doc/chap6_mj.html#X78CEF1F27ED8D7BB>
that constructs a polycyclic group that is infinite and non-virtually
nilpotent. These groups are of the form O(F) \rtimes U(F), where F is an
algebraic number field and O(F) and U(F) are respectively the maximal order
and unit group of F, as discussed in the chapter on polycyclic group
in the Handbook
of Computational Group Theory
<https://www.crcpress.com/Handbook-of-Computational-Group-Theory/Holt-Eick-OBrien/p/book/9781584883722>
.

Is there another method for constructing infinite, non-nilpotent polycyclic
groups that doesn't rely on the split extension method above? And can this
method be implemented in GAP, say via a finite presentation?

Thanks in advance,
Jonathan