ATLAS: Monster group M

The following information is available for M:

• Standard generators.
• Black box algorithms.
• Representations.
• Maximal subgroups, with representations thereof.

Standard generators

Black box algorithms

Finding generators

To find standard generators for M:

• Find an element of order 34, 38, 50, 54, 62, 68, 94, 104 or 110. This powers up to x in class 2A.
  [The probability of success at each attempt is 56542883129 in 363405814200 (about 1 in 6).]
• Find an element of order 9, 18, 27, 36, 45 or 54. This powers up to y in class 3B.
  [The probability of success at each attempt is 3164 in 59049 (about 1 in 19).]
• Find a conjugate a of x and a conjugate b of y such that ab has order 29.
  [The probability of success at each attempt is 1632586752 in 111045174695 (about 1 in 68).]
• Now standard generators of M have been obtained.

Checking generators

To check that elements x and y of M are standard generators:

• Check o(x) = 2
• Check o(y) = 3
• Check o(xy) = 29
• Let u = (xy)⁴(xyy)²
• Check o(u) = 50
• Check o(xu²⁵)
• Check o(xy^xyxyxyxyxyy) = 34

Representations

Update 29th May 1997: generators for the Monster in its 196882-dimensional representation over GF(2) now exist on computer. They are in a special format, requiring special programmes to use them, so they are not being made generally available at this time.

Update 19th November 1997: standard generators can now be made as 196882 × 196882 matrices over GF(2), for the cost of a few days of CPU time, but we do not have enough room to write down the answer.

Update 15th December 1998: standard generators have now been made as 196882 × 196882 matrices over GF(2) — this took about 8 hours CPU time on a pentium machine. They have been multiplied together, using most of the computing resources of Lehrstuhl D für Mathematik, RWTH Aachen, for about 45 hours — completed 05:40 on December 14th.

Update January 2000: generators for the Monster in its 196882-dimensional representation over GF(3) now exist on computer, constructed by Beth Holmes. They are in a special format, requiring special programmes to use them, so they are not being made generally available at this time.

Update October 2000: generators for the Monster in its 196883-dimensional representation over GF(7) now exist on computer. They are in a special format, requiring special programmes to use them, so they are not being made generally available at this time.

Please enquire by e-mail if you require any further information.

Maximal subgroups of the Monster

1. 2.B, order 8 309 562 962 452 852 382 355 161 088 000 000.
2. 2¹⁺²⁴.Co₁, order 139 511 839 126 336 328 171 520 000.
3. 3.Fi₂₄, order 7 531 234 255 143 970 327 756 800. Available as
   • Permutations on 920808 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
   • Dimension 1566 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
4. 2².²E₆(2):S₃, order 1 836 779 512 410 596 494 540 800. Available as
   • Dimension 1708 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
5. 2¹⁰⁺¹⁶.O₁₀⁺(2), order 1 577 011 055 923 770 163 200.
6. 2^2+11+22.(M₂₄ × S₃), order 50 472 333 605 150 392 320. Available as
   • Permutations on 294912 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
7. 3¹⁺¹².2Suz.2, order 2 859 230 155 080 499 200. This group is a quotient of the split extension 3¹⁺¹²:6Suz.2 by a normal subgroup of order 3. We give three generators for this split extension, and the fourth element is a generator for the subgroup of order 3 which has to be factored out. It is now available as a faithful representation of dimension 78; no subgroup of order 3 needs to be factored out. NB: Word for A in Magma files corrected [up to inversion] on 24/8/04. Available as
   • Dimension 38 over GF(3): C , D , E and x (Meataxe); C , D , E and x (Meataxe binary); C , D , E and x (GAP); C, D, E and x (Magma).
   • Dimension 78 over GF(3): C , D and E (Meataxe); C , D and E (Meataxe binary); C , D and E (GAP); C, D and E (Magma).
8. 2^5+10+20.(S₃ × L₅(2)), order 2 061 452 360 684 666 880.
9. S₃ × Th, order 544 475 663 327 232 000. Available as
   • Dimension 250 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
10. 2^3+6+12+18.(3S₆ × L₃(2)), order 199 495 389 743 677 440. Available as
    • Permutations on 1032192 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
11. 3⁸.O₈^-(3).2₃, order 133 214 132 225 341 440. Available as
    • Permutations on 805896 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 204 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
12. (D₁₀ × HN).2, order 5 460 618 240 000 000. Available as
    • Dimension 135 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
13. (3²:2 × O₈⁺(3)).S₄, order 2 139 341 679 820 800. Available as
    • Permutations on 3369 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
14. 3²⁺⁵⁺¹⁰.(M₁₁ × 2S₄), order 49 093 924 366 080. Available as
    • Permutations on 34992 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 69984 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
15. 3^3+2+6+6:(L₃(3) × SD₁₆), order 11 604 018 486 528. Available as
    • Permutations on 85293 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 113724 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 227448 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    These are representations of proper images of 3^3+2+6+6:(L₃(3) × SD₁₆), with generators being images of what we have used above.
    • Permutations on 108 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma). — representing 3⁶:(L₃(3) × SD₁₆).
    • Permutations on 1404 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma). — representing 3⁶⁺⁶:(L₃(3) × SD₁₆).
    • Permutations on 6561 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma). — representing 3²⁺⁶⁺⁶:(L₃(3) × SD₁₆).
16. 5¹⁺⁶:2J₂:4, order 378 000 000 000. Available as
    • Permutations on 78125 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 8 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
17. (7:3 × He):2, order 169 276 262 400. Available as
    • Permutations on 2065 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
18. (A₅ × A₁₂):2, order 28 740 096 000. Available as
    • Permutations on 17 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
19. 5³⁺³.(2 × L₃(5)), order 11 625 000 000. Available as
    • Permutations on 7750 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 46500 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 96875 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 46 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    The group below is also of shape 5³⁺³.(2 × L₃(5)), but is not isomorphic to a subgroup of the Monster. [This group is not of shape 5³⁺³:(2 × L₃(5)) or (5³ × 5³).(2 × L₃(5)) either.] This group has been placed here for purposes of comparison.
    • Permutations on 7750 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 46500 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 96875 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 46 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
20. (A₆ × A₆ × A₆).(2 × S₄), order 2 239 488 000. Available as
    • Permutations on 30 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
21. (A₅ × U₃(8):3₁):2, order 1 985 679 360. Available as
    • Permutations on 518 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 3653 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 28 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
22. 5²⁺²⁺⁴:(S₃ × GL₂(5)), order 1 125 000 000. Available as
    • Permutations on 750 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 15625 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
23. (L₃(2) × S₄(4):2).2, order 658 022 400. Available as
    • Permutations on 184 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 524 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 22 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
24. 7¹⁺⁴:(3 × 2S₇), order 508 243 680. Available as
    • Permutations on 16807 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 6 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
25. (5²:[2⁴] × U₃(5)).S₃, order 302 400 000. Available as
    • Permutations on 151 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
26. (L₂(11) × M₁₂):2, order 125 452 800. Available as
    • Permutations on 36 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
27. (A₇ × (A₅ × A₅):2²):2, order 72 576 000. Available as
    Corrected on 12.08.04. (Old permutations generate (A7 × (A5 × A5):4):2.)
    • Permutations on 17 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
28. 5⁴:(3 × 2L₂(25)):2₂, order 58 500 000. Available as
    • Permutations on 625 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 5 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
29. 7²⁺¹⁺²:GL₂(7), order 33 882 912. Available as
    • Permutations on 392 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Permutations on 2401 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 7 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
30. M₁₁ × A₆.2², order 11 404 800. Available as
    • Permutations on 21 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
31. (S₅ × S₅ × S₅):S₃, order 10 368 000. Available as
    • Permutations on 15 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
32. (L₂(11) × L₂(11)):4, order 1 742 400. Available as
    • Permutations on 24 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
33. 13²:2L₂(13).4, order 1 476 384. Available as
    • Permutations on 169 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 3 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
34. (7²:(3 × 2A₄) × L₂(7)).2, order 1 185 408. Available as
    • Permutations on 57 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
35. (13:6 × L₃(3)).2, order 876 096. Available as
    • Permutations on 39 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
36. 13¹⁺²:(3 × 4S₄), order 632 736. Available as
    • Permutations on 2197 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 4 over GF(13): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
37. L₂(71), order 178 920. Available as
    • Permutations on 72 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
38. L₂(59), order 102 660. Available as
    • Permutations on 60 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
39. 11²:(5 × 2A₅), order 72 600. Available as
    • Permutations on 121 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 3 over GF(11): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
40. L₂(29):2, order 24 360. Available as
    • Permutations on 30 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
41. 7²:SL₂(7), order 16 464. Available as
    • Permutations on 49 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
    • Dimension 3 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
42. L₂(19):2, order 6 840. Available as
    • Permutations on 20 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
43. 41:40, order 1 640. Available as
    • Permutations on 41 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).

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