The quaternionic reflection groups with imprimitive complexifications and the finite subgroups of GL(2,C)

Abstract

We give a transparent development of the quaternionic reflection groups which have imprimitive complexifications, including a description of their reflections and the corresponding normal reflection subgroups. They are seen to be of rank one or rank two, and are obtained from minimal complex reflection groups by the addition of one quaternionic reflection and a complex scalar matrix of finite order. In the case of the rank two groups, we use our techniques to consider the broader question of what are the minimal irreducible subgroups of GL(2,C) with a given collineation group. This leads to a complete classification of the finite irreducible subgroups of GL(2,C) in terms of the minimal groups, with the groups giving rise to quaternionic reflection groups located within it.