University of Isfahan International Journal of Group Theory 2251-7650 3 1 2014 03 01 All simple groups with order from 1 million to 5 million are efficient 17 30 2984 10.22108/ijgt.2014.2984 EN Colin M. Campbell School of Mathematics and Statistics, University of St Andrews George Havas Centre for Discrete Mathematics and Computing, School of Information Technology and Electrical Engineering, The University of Queensland Colin Ramsay Centre for Discrete Mathematics and Computing, School of Information Technology and Electrical Engineering, The University of Queensland Edmund F. Robertson School of Mathematics and Statistics, University of St Andrews Journal Article 2013 03 04 ‎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. https://ijgt.ui.ac.ir/article_2984_1966833cf149fe7e096cd9874914cd5c.pdf