ATLAS: Monster group M
Order = 808017424794512875886459904961710757005754368000000000 = 2^{46}.3^{20}.5^{9}.7^{6}.11^{2}.13^{3}.17.19.23.29.31.41.47.59.71.
Mult = 1.
Out = 1.
The following information is available for M:
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.
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(xu^{25})
 Check o(xy^{xyxyxyxyxyy}) = 34
This algorithm is available in computer readable format:
checker for M.
There are no representations available (yet!).
Update 29th May 1997: generators for the Monster in its
196882dimensional 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
196882dimensional 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
196883dimensional 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 email
if you require any further information.
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:
L_{2}(13), L_{2}(27), Sz(8),
U_{3}(4), U_{3}(8).
 2.B, order
8 309 562 962 452 852 382 355 161 088 000 000.
 2^{1+24}.Co_{1}, order
139 511 839 126 336 328 171 520 000.
 3.Fi_{24}, 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).
 2^{2}.^{2}E_{6}(2):S_{3},
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).
 2^{10+16}.O_{10}^{+}(2), order
1 577 011 055 923 770 163 200.
 2^{2+11+22}.(M_{24} × S_{3}), 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).
 3^{1+12}.2Suz.2, order
2 859 230 155 080 499 200. This group is a quotient
of the split extension 3^{1+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).
 2^{5+10+20}.(S_{3} × L_{5}(2)), order
2 061 452 360 684 666 880.
 S_{3} × 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).
 2^{3+6+12+18}.(3S_{6} × L_{3}(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).
 3^{8}.O_{8}^{}(3).2_{3}, 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).
 (D_{10} × 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).
 (3^{2}:2 × O_{8}^{+}(3)).S_{4}, 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).
 3^{2+5+10}.(M_{11} × 2S_{4}), 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).
 3^{3+2+6+6}:(L_{3}(3) × SD_{16}), 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}:(L_{3}(3) × SD_{16}), 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 3^{6}:(L_{3}(3) × SD_{16}).

Permutations on 1404 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
— representing 3^{6+6}:(L_{3}(3) × SD_{16}).

Permutations on 6561 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
— representing 3^{2+6+6}:(L_{3}(3) × SD_{16}).
 5^{1+6}:2J_{2}: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).
 (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).
 (A_{5} × A_{12}):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).
 5^{3+3}.(2 × L_{3}(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 5^{3+3}.(2 × L_{3}(5)),
but is not isomorphic to a subgroup of the Monster.
[This group is not of shape 5^{3+3}:(2 × L_{3}(5)) or (5^{3} × 5^{3}).(2 × L_{3}(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).
 (A_{6} × A_{6} × A_{6}).(2 × S_{4}), 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).
 (A_{5} × U_{3}(8):3_{1}):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).
 5^{2+2+4}:(S_{3} × GL_{2}(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).
 (L_{3}(2) × S_{4}(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).
 7^{1+4}:(3 × 2S_{7}), 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).
 (5^{2}:[2^{4}] × U_{3}(5)).S_{3}, 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).
 (L_{2}(11) × M_{12}):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).
 (A_{7} × (A_{5} × A_{5}):2^{2}):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).
 5^{4}:(3 × 2L_{2}(25)):2_{2}, 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).
 7^{2+1+2}:GL_{2}(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).
 M_{11} × A_{6}.2^{2}, 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).
 (S_{5} × S_{5} × S_{5}):S_{3}, 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).
 (L_{2}(11) × L_{2}(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).
 13^{2}:2L_{2}(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).
 (7^{2}:(3 × 2A_{4}) × L_{2}(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).
 (13:6 × L_{3}(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).
 13^{1+2}:(3 × 4S_{4}), 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).
 L_{2}(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).
 L_{2}(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).
 11^{2}:(5 × 2A_{5}), 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).
 L_{2}(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).
 7^{2}:SL_{2}(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).
 L_{2}(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).
 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).
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old M page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 13th April 1999.
Last updated 15.11.06 by JNB.
Information checked to
Level 1 on 14.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.