# ATLAS: Monster group M

Order = 808017424794512875886459904961710757005754368000000000 = 246.320.59.76.112.133.17.19.23.29.31.41.47.59.71.
Mult = 1.
Out = 1.

The following information is available for M:

### Standard generators

Standard generators of the Monster group M are a and b where a is in class 2A, b is in class 3B and ab has order 29.

### 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.
This algorithm is available in computer readable format: finder for M.

#### 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(xyxyxyxyxyxyy) = 34
This algorithm is available in computer readable format: checker for M.

### Representations

There are no representations available (yet!).

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

Here we shall include some representations of some maximal subgroups of the Monster, to facilitate calculations in these subgroups. There are 43 classes of maximal subgroups known so far. Any possible maximal subgroup which is not listed here has socle isomorphic to one of the following simple groups: L2(13), L2(27), Sz(8), U3(4), U3(8).
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
4. 22.2E6(2):S3, order 1 836 779 512 410 596 494 540 800. Available as
5. 210+16.O10+(2), order 1 577 011 055 923 770 163 200.
6. 22+11+22.(M24 × S3), order 50 472 333 605 150 392 320. Available as
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
8. 25+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
10. 23+6+12+18.(3S6 × L3(2)), order 199 495 389 743 677 440. Available as
11. 38.O8-(3).23, order 133 214 132 225 341 440. Available as
12. (D10 × HN).2, order 5 460 618 240 000 000. Available as
13. (32:2 × O8+(3)).S4, order 2 139 341 679 820 800. Available as
14. 32+5+10.(M11 × 2S4), order 49 093 924 366 080. Available as
15. 33+2+6+6:(L3(3) × SD16), order 11 604 018 486 528. Available as
These are representations of proper images of 33+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
17. (7:3 × He):2, order 169 276 262 400. Available as
18. (A5 × A12):2, order 28 740 096 000. Available as
19. 53+3.(2 × L3(5)), order 11 625 000 000. Available as
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.
20. (A6 × A6 × A6).(2 × S4), order 2 239 488 000. Available as
21. (A5 × U3(8):31):2, order 1 985 679 360. Available as
22. 52+2+4:(S3 × GL2(5)), order 1 125 000 000. Available as
23. (L3(2) × S4(4):2).2, order 658 022 400. Available as
24. 71+4:(3 × 2S7), order 508 243 680. Available as
25. (52:[24] × U3(5)).S3, order 302 400 000. Available as
26. (L2(11) × M12):2, order 125 452 800. Available as
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.)
28. 54:(3 × 2L2(25)):22, order 58 500 000. Available as
29. 72+1+2:GL2(7), order 33 882 912. Available as
30. M11 × A6.22, order 11 404 800. Available as
31. (S5 × S5 × S5):S3, order 10 368 000. Available as
32. (L2(11) × L2(11)):4, order 1 742 400. Available as
33. 132:2L2(13).4, order 1 476 384. Available as
34. (72:(3 × 2A4) × L2(7)).2, order 1 185 408. Available as
35. (13:6 × L3(3)).2, order 876 096. Available as
36. 131+2:(3 × 4S4), order 632 736. Available as
37. L2(71), order 178 920. Available as
38. L2(59), order 102 660. Available as
39. 112:(5 × 2A5), order 72 600. Available as
40. L2(29):2, order 24 360. Available as
41. 72:SL2(7), order 16 464. Available as
42. L2(19):2, order 6 840. Available as
43. 41:40, order 1 640. Available as

Go to main ATLAS (version 2.0) page.