Nice Efficient Presentations for All Small Simple Groups and Their Covers

Prior to this paper all small simple groups were known to be efficient but the status of four of their covering groups was unknown. We produce nice efficient presentations for all of these groups, resolving the previously unknown cases. We give presentations that are better than available before in terms of length and in terms of computational properties. In many cases our presentations have minimal possible length. Our results are based on major amounts of computation. We make substantial use of systems for computational group theory and, in particular, of computer implementations of coset enumeration. To assist in reducing the number of relators we provide theorems which enable the amalgamation of power relations in certain presentations. We conclude with a selection of unsolved problems about efficient presentations for simple groups and their covers.