Systems of imprimitivity for rank two quaternionic reflection groups

TL;DR

This paper revises the classification of rank two quaternionic reflection groups, identifying missing groups, establishing isomorphisms, and revealing that some primitive complex groups have infinitely many systems of imprimitivity in quaternionic form.

Contribution

It provides a corrected and extended enumeration of imprimitive rank two quaternionic reflection groups and explores their systems of imprimitivity and isomorphisms.

Findings

Added missing groups to the classification.

Established isomorphisms between groups in existing tables.

Discovered primitive complex groups with infinitely many systems of imprimitivity.

Abstract

We revise the enumeration of the imprimitive rank two quaternionic reflection groups, adding missing groups and establishing isomorphisms between groups in the published tables. The isomorphisms are obtained as a consequence of the determination of the reflection groups with more than one system of imprimitivity. We find that there are primitive complex reflection groups which have infinitely many systems of imprimitivity when represented as quaternionic reflection groups.