ATLAS: Baby Monster group B

Order = 4154781481226426191177580544000000 = 2⁴¹.3¹³.5⁶.7².11.13.17.19.23.31.47.
Mult = 2.
Out = 1.

The following information is available for B:

Standard generators.
Black box algorithms.
Representations.
Maximal subgroups.
Conjugacy classes.

Standard generators

Standard generators of the Baby Monster are a and b where a is in class 2C, b is in class 3A, ab has order 55 and ababababbababbabb has order 23.

Black box algorithms

Finding generators

To find standard generators for B:

Find any element of order 52. This powers up to a 2C-element, x say.
Find any element of order 21, 33, 39, 48, 66. This powers up to a 3A-element, y say.
Find a conjugate a of x and a conjugate b of y, whose product has order 55.
If ababb has order 35, go back to previous step.
Otherwise, ababb has order 40. If (ab)²(ababb)²abb has order 23, you have finished, while if not, the order will be 31, and you should replace b by its inverse.

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

Checking generators

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

Check o(x) = 2
Check o(y) = 3
Check o(xy) = 55
Check o(xyxyxyxyyxyxyyxyy) = 23
Let u = (xyxyy)²(xy)²(xyxyy)²
Check o(u) = 52
Check o(xu²⁶) = 35
Let v = (xy)³(xyyxy)²xy(xyxyy)²xyy
Check o(v) = 38
Check o(v¹⁹y^x) = 8

This algorithm is available in computer readable format: checker for B.

Representations

The representations of B available are:

Dimension 4370 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
Dimension 4371 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).
Dimension 4371 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP), a and b (Magma).

Maximal subgroups

The maximal subgroups include the following, in decreasing order of size. (This list is now (since 15th January 1997) believed to be complete.)

2.²E₆(2):2, with generators (a((ab)¹⁴abb)¹⁹)³, (ab)¹⁴abb. To make standard generators (modulo the central involution), make the following words in the above generators (x, y): bla bla.
Order 306 129 918 735 099 415 756 800.
Index 13 571 955 000.
2¹⁺²².Co₂, with generators (a((ab)⁴ab²)²⁴)², (ab)⁴ab². To make standard generators (modulo the normal 2-group), make the following words in the above generators (x, y): ((xy)^3yxy)^5, (xyy(xyxyy)^3)^-1(y(xyxyy)^2)^4xyy(xyxyy)^3.
Order 354 883 595 661 213 696 000.
Index 11 707 448 673 375.
Fi₂₃, with standard generators (ab)^-10(ababb)^20(ab)^10, (abb)^-9(abababbab)^4(abb)^9.
Order 4 089 470 473 293 004 800.
Index 1 015 970 529 280 000.
2⁹⁺¹⁶.S₈(2), with generators [(abababb)^10(ab[(abababb)^10,ab]^17abb[(abababb)^10,abb]^16)^20,ab]^12, ab[(abababb)^10,ab]^17.
Order 1 589 728 887 019 929 600.
Index 2 613 515 747 968 125.
Th, with standard generators (ab)^-13(abababb)¹⁰(ab)¹³, (abb)^-17b(abb)¹⁷.
Order 90 745 943 887 872 000.
Index 45 784 762 417 152 000.
(2² × F₄(2)):2, with generators (a(abababbababb)^9)^13, ab((abababbababb)^9(ab)^2(abababbababb)^9ab)^5.
Order 26 489 012 826 931 200.
Index 156 849 238 149 120 000.
2^2+10+20.(M₂₂:2 × S₃), with generators bla bla.
Order 22 858 846 741 463 040.
Index 181 758 140 654 146 875.
[2³⁰].L₅(2), with generators bla bla.
Order 10 736 731 045 232 640.
Index 386 968 944 618 506 250.
S₃ × Fi₂₂:2, with generators bla bla.
Order 774 741 019 852 800.
Index 5 362 800 438 804 480 000.
[2³⁵].(S₅ × L₃(2)).
Order 692 692 325 498 880.
Index 5 998 018 641 586 846 875.
HN:2, with generators (ba)^-3a(ba)³, (ab)^-4b(ab)⁴.
Order 546 061 824 000 000.
Index 7 608 628 361 513 926 656.
O₈⁺(3):S₄.
Order 118 852 315 545 600.
Index 34 957 513 971 466 240 000.
3¹⁺⁸.2¹⁺⁶.U₄(2).2, with generators here.
Order 130 606 940 160.
Index 31 811 337 714 034 278 400 000.
(3²:D₈ × U₄(3).2.2).2.
Order 1 881 169 920.
Index 2 208 615 732 717 237 043 200 000.
5:4 × HS:2.
Order 1 774 080 000.
Index 2 341 935 809 673 986 624 716 800.
S₄ × ²F₄(2), with generators here.
Order 862 617 600.
Index 4 816 481 232 502 590 013 440 000.
[3¹¹].(S₄ × 2S₄).
Order 204 073 344.
Index 20 359 256 136 981 938 176 000 000.
S₅ × M₂₂:2.
Order 106 444 800.
Index 39 032 263 494 566 443 745 280 000.
(S₆ × L₃(4):2).2.
Order 58 060 800.
Index 71 559 149 740 038 480 199 680 000.
5³.L₃(5).
Order 46 500 000.
Index 89 350 139 381 213 466 476 937 216.
5¹⁺⁴.2¹⁺⁴.A₅.4, with generators here.
Order 24 000 000.
Index 173 115 895 051 101 091 299 065 856.
(S₆ × S₆).4.
Order 2 073 600.
Index 2 003 656 192 721 077 445 591 040 000.
5²:4S₄ × S₅.
Order 288 000.
Index 14 426 324 587 591 757 608 255 488 000.
L₂(49).2c.
Order 117 600.
Index 35 329 774 500 224 712 510 013 440 000.
L₂(31).
Order 14 880.
Index 279 219 185 566 292 082 740 428 800 000.
M₁₁.
Order 7 920.
Index 524 593 621 366 973 003 936 563 200 000.
L₃(3).
Order 5 616.
Index 739 811 517 312 397 826 064 384 000 000.
L₂(17):2.
Order 4 896.
Index 848 607 328 681 868 094 603 264 000 000.
L₂(11):2.
Order 1 320.
Index 3 147 561 728 201 838 023 619 379 200 000.
47:23, with generators here. A Magma version is here. This subgroup is generated by the 47-element ababab²abababab²abab², together with an element of order 23 that conjugates it to its 16th power.
Order 1 081.
Index 3 843 461 129 719 173 164 826 624 000 000.

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements, for example by running this program afterwards.

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Version 2.0 created on 20th December 2000.
Last updated 12.05.06 by JNB.
Information checked to Level 0 on 20.12.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.