All Simple Groups With Order From 1 Million To 5 Million Are Efficient

There is much interest in finding short presentations for the finite simple groups. Indeed it has been suggested that all these groups are efficient in a technical sense. In previous papers we produced nice efficient presentations for all except one of the simple groups with order less than one million. Here we show that all simple groups with order between 1 million and 5 million are efficient by giving efficient presentations for all of them. Apart from some linear groups these results are all new. We also show that some covering groups and some larger simple groups are efficient We make substantial use of systems for computational group theory and, in particular, of computer implementations of coset enumeration to find and verify our presentations.