There is a large ecosystem of mathematical software systems. Individually, these are optimized for particular domains and functionalities, and together they cover many needs of practical and theoretical mathematics. However, each system specializes on one particular area, and it remains very difficult to solve problems that need to involve multiple systems. Some integrations exist, but the are ad-hoc and have scalability and maintainability issues. In particular, there is not yet an interoperability layer that combines the various systems into a virtual research environment (VRE) for mathematics. The OpenDreamKit project aims at building a toolkit for such VREs. It suggests using a central system-agnostic formalization of mathematics (Math-in-the-Middle, MitM) as the needed interoperability layer. In this paper, we report on a case study that instantiates the MitM paradigm the systems GAP, SageMath, and Singular to perform computation in group and ring theory. Our work involves massive practical efforts, including a novel formalization of computational group theory, improvements to the involved software systems, and a novel mediating system that sits at the center of a star-shaped integration layout between mathematical software systems. © Springer International Publishing AG 2017.