ATLAS: Baby Monster

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

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

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.

Representations

The representations available are
• a and b as 4370 x 4370 matrices over GF(2).
• a and b as 4371 x 4371 matrices over GF(3).
• a and b as 4371 x 4371 matrices over GF(5).

Maximal subgroups

The maximal subgroups include the following, in decreasing order of size. (This list is now (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.

• 2^1+22Co2, with generators (a((ab)^4abb)^24)^2, (ab)^4abb. 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.

• 2^9+16.S8(2).

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.

• (2^2 x F4(2)):2.

Order 26 489 012 826 931 200.
Index 156 849 238 149 120 000.

• 2^2+10+20(M22:2 x S3)

Order 22 858 846 741 463 040.
Index 181 758 140 654 146 875.

• [2^30].L5(2).

Order 10 736 731 045 232 640.
Index 386 968 944 618 506 250.

• S3 x F22:2

Order 774 741 019 852 800.
Index 5 362 800 438 804 480 000.

• [2^35].(S5 x 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.

• 3^1+8.2^1+6.U4(2).2

Order 130 606 940 160.
Index 31 811 337 714 034 278 400 000.

• (3^2D8 x U4(3).2.2).2

Order 1 881 169 920.
Index 2 208 615 732 717 237 043 200 000.

• 5:4 x HS:2

Order 1 774 080 000.
Index 2 341 935 809 673 986 624 716 800.

• S4 x ^2F4(2).

Order 862 617 600.
Index 4 816 481 232 502 590 013 440 000.

• [3^11](S4 x 2S4)

Order 204 073 344.
Index 20 359 256 136 981 938 176 000 000.

• S5 x M22:2

Order 106 444 800.
Index 39 032 263 494 566 443 745 280 000.

• (S6 x L3(4):2).2

Order 58 060 800.
Index 71 559 149 740 038 480 199 680 000.

• 5^3.L3(5).

Order 46 500 000.
Index 89 350 139 381 213 466 476 937 216.

• 5^1+4.2^1+4.A5.4

Order 24 000 000.
Index 173 115 895 051 101 091 299 065 856.

• (S6 x S6).4

Order 2 073 600.
Index 2 003 656 192 721 077 445 591 040 000.

• 5^2:4S4 x S5.

Order 288 000.
Index 14 426 324 587 591 757 608 255 488 000.

• L2(49).2

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

Order 1 081.
Index 3 843 461 129 719 173 164 826 624 000 000.

Return to main ATLAS page. Last updated 15.11.99

R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk