All simple groups with order from 1 million to 5 million are efficient

Abstract

‎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.