I am looking to do a case-by-case check of a conjecture I have about Shephard groups, which are a subclass of complex reflection groups. These were classified by Shephard and Todd and there is one infinite class and 34 exceptional cases. I have shown the result for the groups $G(r,1,n)$. To finish checking my conjecture, I need to know the "roots" associated to each of the exceptional groups (well, I guess just the Shephard groups).
Let me elaborate: Each complex reflection group is generated by generalized reflections, which are elements of $GL(V)$ with finite order that fix a hyperplane point-wise. I need to know what these hyperplanes are.
I'm looking for a list along the lines of what occurs in the "Plates" of Bourbaki's book on Lie Groups, or in Section 2.10 of Humphrey's book on real reflection groups.
Thanks.