By American Mathematical Society

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G is the group of symmetries of determinant 1. 2 Structure Character 10 6 Orthogona1 (2) 1a+5a N(5AB) 1a+4a+S a N(3A) Linear (5) isotropic point Linear (4) Orthogona1 (4) point point isotropic point °2 (4), L1(16) minus line 0"2(4), base plus line duad Orthogona1 (5) Icosahedral base base point isotropic point pentad axis non-isotropic point °2 (5), L1(25) minus point triad axis @ @ @ @ @ 4 3 5 5 A A A A A A A A 2A 3A 5A B* fus ind + @ @ 623 A A AB A A AB 2B 4A 6A ++ 3 -1 o -b5 ° x, + 3 -1 x. + 4 x, + 5 ind 1 2 ° -1 4 3 6 2 x, 2 ° -1 ° -1 x.

O 0 0 0 o .. -b7 ,0 y16 *3 o ! 2 3 3. 22 Character Abstract Alternating 6 A5 35 D20 5:4 10:4 la+5a N(2A,3A,5A) point 60 6 35 la+5b N(2A,3B,5A) total 36 10 A 5 32 :4 2 3 : DS 32 :S 2 3 : QS 32 : (2 4 ] la+9a N(3 2 ) = N( 3A2B2) bisection 24 15 34 34 x 2 D16 S:2 (2 5 ] la+5a+9a N(2A 2 ), C(2B) duad 24 15 34 34 x 2 la+5b+9a N(2A 2 ), C(2C) syntheme Order Index 60 Linear (9) [4J Specifications Orthogonal (9) Orthogonal (3) Orthogonal (2) 3ymplectic Mathieu Hexacode icosahedral minus plane 0q(2) , 32 (4) total coordinate icosahedral minus plane 0q(2), 32 (4) total weight 6 word point isotropic point isotropic point plus plane 01;(2) point L2 (3) °3(3) , base non-isotropic point isotropic point point tetrad weight 4 word L2 (3) ° 3 (3) , base non-isotropic point isotropic line isotropic line tetrad 2 weight 6 words @ 360 power p' part ind 1A p @ @ @ @ @ @ 8 A 9 A 9 A 4 A 5 A 5 A A A A A A A 3A 3B 2A 4A 5A @ B* rus ind @ + X2 + 5 2 -1 -1 0 0 ++ X, + 5 -1 2 -1 0 0 ++ -1 3 + 8 0 ~1 -1 0 -b5 * + 0 0 Xs + 8 0 -1 -1 0 * -b5 X.

2), [the Monster], and finally the Aachen 'CAS' team led by J. Neubtiser and H. Pahlings both for many original tables and for improvements, extensions, and corrections to many others. Among our group-theoretical colleagues at Cambridge who have used the A lr ILA§ and contributed tables, corrections, improvements, or criticism are David Benson, Patrick Brooke, Mike Guy, David Jackson, Gordon James, Peter Kleidman, Martin Liebeck, Nick Patterson, Larissa Queen, Alex Ryba, Jan Saxl, Peter Smith, and finally John Thompson, who has acted as our friend and mentor throughout.

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