Jacob Hirbawi asked about the possibility to calculate with the
elements of an infinite matrix group over the rationals. That is
indeed possible as well as the calculation with matrices over
cyclotomic fields, it is just that most of the general group functions
cannot be used. Frank Celler is answering that in more detail.

However,in a PS you say that this has to do with a number of problems
in crystallography and since in the past I have been working on
crystallographic groups making heavy use of computer methods (e.g. for
the classification of the four-dimensional point and space groups), I
am curious to know what you are after. possibly we or others might
even have some suggestions how to use GAP for the problems.

Joachim Neubueser