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

Porting notes

Porting incomplete.

Standard generators

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

Black box algorithms

Finding generators

Group Algorithm File
M Download

Checking generators (semi-presentations)

Group Semi-presentation File
M 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 29, o(u) = 50, o(au25) = 5, o(ababababababb) = 34; u = (ab)4(abb)2 〉〉 Download

Representations

Maximal subgroups

Maximal subgroups of M

Subgroup Order Index Programs/reps
2.B 8 309 562 962 452 852 382 355 161 088 000 000 97 239 461 142 009 186 000
21+24.Co1 139 511 839 126 336 328 171 520 000 5 791 748 068 511 982 636 944 259 375
3.Fi24 1 836 779 512 410 596 494 540 800 439 909 863 614 532 427 326 210 000 000 Rep: permutations on 920808 points
Rep: 1566-dimensional over GF(2)
22.2E6:S3 1 836 779 512 410 596 494 540 800 439 909 863 614 532 427 326 210 000 000 Rep: 1708-dimensional over GF(2)
210+16.O10+(2) 1 577 011 055 923 770 163 200 512 372 707 698 741 056 749 515 292 734 375
22+11+22.(M24 × S3) 50 472 333 605 150 392 320 16 009 115 629 875 684 006 343 550 944 921 875 Rep: permutations on 294912 points
31+12.2Suz.2 2 859 230 155 080 499 200 282 599 644 298 926 271 851 701 207 040 000 000 Rep: 38-dimensional over GF(3) This group is a quotient of the split extension 3<SUP>1+12</SUP>: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
Rep: 78-dimensional over GF(3)
25+10+20.(S3 × L5(2)) 2 061 452 360 684 666 880 391 965 121 389 536 908 413 379 198 941 796 875
S3 × Th 544 475 663 327 232 000 1 484 028 541 986 258 159 045 049 319 424 000 000 Rep: 250-dimensional over GF(2)
23+6+12+18.(L3(2) × 3S6) 199 495 389 743 677 440 4 050 306 254 358 548 053 604 918 389 065 234 375
38.O8−(3).23 133 214 132 225 341 440 6 065 553 341 050 124 859 256 025 907 200 000 000 Rep: permutations on 805896 points
Rep: 204-dimensional over GF(3)
(D10 × HN).2 5 460 618 240 000 000 147 971 784 380 684 498 443 615 773 616 452 403 200 Rep: 135-dimensional over GF(5)
(32:2 × O8+(3)).S4 2 139 341 679 820 800 377 694 424 605 514 962 329 798 663 208 960 000 000 Rep: permutations on 3369 points
32+5+10.(M11 × 2S4) 49 093 924 366 080 16 458 603 283 969 466 072 643 078 298 009 600 000 000 Rep: permutations on 34992 points
Rep: permutations on 69984 points
33+2+6+6:(L3(3) × SD16) 11 604 018 486 528 69 632 552 355 255 433 384 259 177 414 656 000 000 000 Rep: permutations on 85293 points
Rep: permutations on 113724 points
Rep: permutations on 227448 points
Rep: permutations on 108 points This is a representation of the proper image 3<SUP>6</SUP>:(L<SUB>3</SUB>(3) × SD<SUB>16</SUB>)
Rep: permutations on 1404 points This is a representation of the proper image 3<SUP>6+6</SUP>:(L<SUB>3</SUB>(3) × SD<SUB>16</SUB>)
Rep: permutations on 6561 points This is a representation of the proper image 3<SUP>2+6+6</SUP>:(L<SUB>3</SUB>(3) × SD<SUB>16</SUB>)
51+6:2J2:4 378 000 000 000 2 137 612 234 906 118 719 276 348 954 925 160 732 819 456 Rep: permutations on 78125 points
Rep: 8-dimensional over GF(5)
(7:3 × He):2 169 276 262 400 4 773 365 227 577 903 302 562 875 496 013 496 320 000 000 Rep: permutations on 2065 points
(A5 × A12):2 28 740 096 000 28 114 639 032 330 054 704 286 996 987 125 956 608 000 000 Rep: permutations on 17 points
53+3.(2 × L3(5)) 11 625 000 000 69 506 875 251 140 892 549 372 895 050 469 742 538 129 408 Rep: permutations on 7750 points
Rep: permutations on 46500 points
Rep: permutations on 96875 points
Rep: permutations on 7750 points These generate a subgroup of shape 5<SUP>3+3</SUP>.(2 × L<SUB>3</SUB>(5)), but it is not isomorphic to a subgroup of the Monster (it is not of shape 5<SUP>3+3</SUP>:(2 × L<SUB>3</SUB>(5)) or (5<SUP>3</SUP> × 5<SUP>3</SUP>).(2 × L<SUB>3</SUB>(5)) either). This group has been placed here for purposes of comparison,
Rep: permutations on 46500 points These generate a subgroup of shape 5<SUP>3+3</SUP>.(2 × L<SUB>3</SUB>(5)), but it is not isomorphic to a subgroup of the Monster (it is not of shape 5<SUP>3+3</SUP>:(2 × L<SUB>3</SUB>(5)) or (5<SUP>3</SUP> × 5<SUP>3</SUP>).(2 × L<SUB>3</SUB>(5)) either). This group has been placed here for purposes of comparison,
Rep: permutations on 96875 points These generate a subgroup of shape 5<SUP>3+3</SUP>.(2 × L<SUB>3</SUB>(5)), but it is not isomorphic to a subgroup of the Monster (it is not of shape 5<SUP>3+3</SUP>:(2 × L<SUB>3</SUB>(5)) or (5<SUP>3</SUP> × 5<SUP>3</SUP>).(2 × L<SUB>3</SUB>(5)) either). This group has been placed here for purposes of comparison,
Rep: 46-dimensional over GF(5)
Rep: 46-dimensional over GF(5) These generate a subgroup of shape 5<SUP>3+3</SUP>.(2 × L<SUB>3</SUB>(5)), but it is not isomorphic to a subgroup of the Monster (it is not of shape 5<SUP>3+3</SUP>:(2 × L<SUB>3</SUB>(5)) or (5<SUP>3</SUP> × 5<SUP>3</SUP>).(2 × L<SUB>3</SUB>(5)) either). This group has been placed here for purposes of comparison,
(A6 × A6 × A6).(2 × S4) 2 239 488 000 360 804 534 248 235 702 038 349 794 668 116 443 136 000 000 Rep: permutations on 30 points
(A5 × U3(8):31):2 1 985 679 360 406 922 407 046 882 370 719 943 377 445 244 108 800 000 000 Rep: permutations on 518 points
Rep: permutations on 3653 points
Rep: 28-dimensional over GF(2)
52+2+4:(S3 × GL2(5)) 1 125 000 000 718 237 710 928 455 889 676 853 248 854 854 006 227 337 216 Rep: permutations on 750 points
Rep: permutations on 15625 points
(L3(2) × S4(4):2).2 658 022 400 1 227 948 204 794 415 624 584 299 721 349 471 928 320 000 000 Rep: permutations on 184 points
Rep: permutations on 524 points
Rep: 22-dimensional over GF(2)
71+4:(3 × 2S7) 508 243 680 1 589 822 867 634 109 834 649 512 818 264 086 937 600 000 000 Rep: permutations on 16807 points
Rep: 6-dimensional over GF(7)
(52:[24] × U3(5)).S3 302 400 000 2 672 015 293 632 648 399 095 436 193 656 450 916 024 320 000 Rep: permutations on 151 points
(L2(11) × M12):2 125 452 800 6 440 808 214 679 248 895 891 202 946 141 582 786 560 000 000 Rep: permutations on 36 points
(A7 × (A5 × A5):22):2 72 576 000 11 133 397 056 802 701 662 897 650 806 901 878 816 768 000 000 Rep: permutations on 17 points
54:(3 × 2L2(25)):22 58 500 000 13 812 263 671 701 074 801 477 947 093 362 577 042 833 408 000 Rep: permutations on 625 points
Rep: 5-dimensional over GF(5)
72+1+2:GL2(7) 33 882 912 23 847 343 014 511 647 519 742 692 273 961 304 064 000 000 000 Rep: permutations on 392 points
Rep: permutations on 2401 points
Rep: 7-dimensional over GF(7)
M11 × A6.22 11 404 800 70 848 890 361 471 737 854 803 232 407 557 410 652 160 000 000 Rep: permutations on 21 points
(S5 × S5 × S5):S3 10 368 000 77 933 779 397 618 911 640 283 555 648 313 151 717 376 000 000 Rep: permutations on 15 points
(L2(11) × L2(11)):4 1 742 400 463 738 191 456 905 920 504 166 612 122 193 960 632 320 000 000 Rep: permutations on 24 points
132:2L2(13).4 1 476 384 547 294 894 007 597 532 814 267 768 386 619 441 152 000 000 000 Rep: permutations on 169 points
Rep: 3-dimensional over GF(13)
(72:(3 × 2A4) × L2(7)).2 1 185 408 681 636 554 498 124 591 605 978 620 830 727 274 496 000 000 000 Rep: permutations on 57 points
(13:6 × L3(3)).2 876 096 922 293 247 309 099 546 038 858 646 725 599 428 608 000 000 000 Rep: permutations on 39 points
131+2:(3 × 4S4) 632 736 1 277 021 419 351 060 909 899 958 126 235 445 362 688 000 000 000 Rep: permutations on 2197 points
Rep: 4-dimensional over GF(13)
L2(71) 178 920 4 516 082 186 421 377 575 935 948 496 320 762 111 590 400 000 000 Rep: permutations on 72 points
L2(59) 102 660 7 870 810 683 757 187 569 515 487 092 944 776 514 764 800 000 000 Rep: permutations on 60 points
112:(5 × 2A5) 72 600 11 129 716 594 965 742 092 099 998 690 932 655 055 175 680 000 000 Rep: permutations on 121 points
Rep: 3-dimensional over GF(11)
L2(29):2 24 360 33 169 845 024 405 290 471 529 552 748 838 701 026 508 800 000 000 Rep: permutations on 30 points
72:SL2(7) 16 464 49 077 831 923 864 970 595 630 460 699 812 363 763 712 000 000 000 Rep: permutations on 49 points
Rep: 3-dimensional over GF(7)
L2(19):2 6 840 118 131 202 455 338 139 749 482 442 245 864 145 761 075 200 000 000 Rep: permutations on 20 points
41:40 1 640 492 693 551 703 971 265 784 426 771 318 116 315 247 411 200 000 000 Rep: permutations on 41 points

Conjugacy classes

Conjugacy classes of M

Conjugacy class Centraliser order Power up Class rep(s)
1A 808 017 424 794 512 875 886 459 904 961 710 757 005 754 368 000 000 000
2A 8 309 562 962 452 852 382 355 161 088 000 000 4B 6A 6D 8C 10A 10C 12C 12G 14A 18A 18B 20B 22A 24G 26A 28A 30B 30F 34A 36C 38A 40A 42A 46C 46D 50A 52A 54A 60A 62A 62B 66A 68A 70A 78A 84A 94A 94B 104A 104B 110A
2B 139 511 839 126 336 328 171 520 000 4A 4C 4D 6B 6C 6E 6F 8A 8B 8D 8E 8F 10B 10D 10E 12A 12B 12D 12E 12F 12H 12I 12J 14B 14C 16A 16B 16C 18C 18D 18E 20A 20C 20D 20E 20F 22B 24A 24B 24C 24D 24E 24F 24H 24I 24J 26B 28B 28C 28D 30A 30C 30D 30E 30G 32A 32B 36A 36B 36D 40B 40C 40D 42B 42C 42D 44A 44B 46A 46B 48A 52B 56A 56B 56C 60B 60C 60D 60E 60F 66B 70B 78B 78C 84B 84C 88A 88B 92A 92B
3A 3 765 617 127 571 985 163 878 400 6A 6C 12A 12C 12E 15A 21A 21B 24A 24B 24D 30B 30C 33B 39A 42A 42D 48A 51A 60A 60B 66A 69A 69B 78A 84A 87A 87B 105A
3B 1 429 615 077 540 249 600 6B 6D 6E 9A 9B 12B 12F 12G 12H 12I 15B 15C 18A 18B 18C 18D 18E 21D 24C 24F 24G 24H 24I 27A 27B 30A 30D 30F 30G 33A 36A 36B 36C 36D 39C 39D 42B 45A 54A 60C 60D 60E 66B 78B 78C 84B
3C 272 237 831 663 616 000 6F 12D 12J 15D 21C 24E 24J 30E 39B 42C 57A 60F 84C 93A 93B
4A 8 317 584 273 309 696 000 8B 12A 12B 12D 20A 20C 24A 24E 28B 36A 36B 40B 44A 44B 60B 60C 84C 88A 88B 92A 92B
4B 26 489 012 826 931 200 8C 12C 12G 20B 24G 28A 36C 40A 52A 60A 68A 84A 104A 104B
4C 48 704 929 136 640 8A 8D 8E 12E 12H 12I 16A 16B 16C 20F 24B 24C 24D 24H 24I 28C 32A 32B 36D 40C 40D 48A 56A 60D
4D 8 244 323 942 400 8F 12F 12J 20D 20E 24F 24J 28D 52B 56B 56C 60E 60F 84B
5A 1 365 154 560 000 000 10A 10B 15A 15B 20A 20B 20D 30B 30C 30D 35A 40A 40B 45A 55A 60A 60B 60E 70A 95A 95B 105A 110A
5B 94 500 000 000 10C 10D 10E 15C 15D 20C 20E 20F 25A 30A 30E 30F 30G 35B 40C 40D 50A 60C 60D 60F 70B
6A 774 741 019 852 800 12C 30B 42A 60A 66A 78A 84A
6B 2 690 072 985 600 12F 24F 30A 30D 42B 60E 66B 78B 78C 84B
6C 481 579 499 520 12A 12E 24A 24B 24D 30C 42D 48A 60B
6D 130 606 940 160 12G 18A 18B 24G 30F 36C 54A
6E 1 612 431 360 12B 12H 12I 18C 18D 18E 24C 24H 24I 30G 36A 36B 36D 60C 60D
6F 278 691 840 12D 12J 24E 24J 30E 42C 60F 84C
7A 28 212 710 400 14A 14B 21A 21C 28A 28B 28C 35A 42A 42C 56A 70A 84A 84C 105A 119A 119B
7B 84 707 280 14C 21B 21D 28D 35B 42B 42D 56B 56C 70B 84B
8A 792 723 456 16A 24B 24C 48A 56A
8B 778 567 680 24A 24E 40B 88A 88B
8C 143 769 600 24G 40A 104A 104B
8D 23 592 960 24D 24H 40C 40D
8E 12 582 912 16B 16C 24I 32A 32B
8F 3 096 576 24F 24J 56B 56C
9A 56 687 040 18B 18C 36A 36C 45A
9B 2 834 352 18A 18D 18E 27A 27B 36B 36D 54A
10A 887 040 000 20B 30B 40A 60A 70A 110A
10B 18 432 000 20A 20D 30C 30D 40B 60B 60E
10C 12 000 000 30F 50A
10D 6 048 000 20E 30A 30E 60F 70B
10E 480 000 20C 20F 30G 40C 40D 60C 60D
11A 1 045 440 22A 22B 33A 33B 44A 44B 55A 66A 66B 88A 88B 110A
12A 119 439 360 24A 60B
12B 22 394 880 36A 36B 60C
12C 17 418 240 60A 84A
12D 1 161 216 24E 84C
12E 884 736 24B 24D 48A
12F 483 840 24F 60E 84B
12G 373 248 24G 36C
12H 276 480 24H 60D
12I 82 944 24C 24I 36D
12J 23 040 24J 60F
13A 73 008 26A 39A 39B 52A 78A 104A 104B
13B 52 728 26B 39C 39D 52B 78B 78C
14A 1 128 960 28A 42A 70A 84A
14B 150 528 28B 28C 42C 56A 84C
14C 35 280 28D 42B 42D 56B 56C 70B 84B
15A 2 721 600 30B 30C 60A 60B 105A
15B 145 800 30D 45A 60E
15C 10 800 30A 30F 30G 60C 60D
15D 9 000 30E 60F
16A 12 288 48A
16B 8 192 32A
16C 8 192 32B
17A 2 856 34A 51A 68A 119A 119B
18A 34 992 54A
18B 23 328 36C
18C 15 552 36A
18D 3 888 36B 36D
18E 3 888
19A 1 140 38A 57A 95A 95B
20A 76 800 40B 60B
20B 28 800 40A 60A
20C 24 000 60C
20D 19 200 60E
20E 1 200 60F
20F 960 40C 40D 60D
21A 52 920 42A 84A 105A
21B 6 174 42D
21C 3 528 42C 84C
21D 504 42B 84B
22A 2 640 66A 110A
22B 2 112 44A 44B 66B 88A 88B
23A 552 23B5 46A 46B 46C 46D 69A 69B 92A 92B
23B 552 23A5 46A 46B 46C 46D 69A 69B 92A 92B
24A 6 912
24B 4 608 48A
24C 3 456
24D 2 304
24E 1 152
24F 864
24G 864
24H 576
24I 384
24J 288
25A 250 50A
26A 624 52A 78A 104A 104B
26B 312 52B 78B 78C
27A 486 54A
27B 243
28A 4 704 84A
28B 2 688 84C
28C 896 56A
28D 168 56B 56C 84B
29A 87 87A 87B
30A 10 800
30B 7 200 60A
30C 2 880 60B
30D 1 800 60E
30E 360 60F
30F 240
30G 240 60C 60D
31A 186 31B3 62A 62B 93A 93B
31B 186 31A3 62A 62B 93A 93B
32A 128
32B 128
33A 594 66B
33B 396 66A
34A 136 68A
35A 2 100 70A 105A
35B 70 70B
36A 1 296
36B 648
36C 216
36D 72
38A 76
39A 702 78A
39B 117
39C 78 39D7 78B 78C
39D 78 39C7 78B 78C
40A 400
40B 320
40C 80 40D11
40D 80 40C11
41A 41
42A 504 84A
42B 504 84B
42C 168 84C
42D 126
44A 352 44B7 88A 88B
44B 352 44A7 88A 88B
45A 135
46A 184 46B5 92A 92B
46B 184 46A5 92A 92B
46C 92 46D5
46D 92 46C5
47A 94 47B5 94A 94B
47B 94 47A5 94A 94B
48A 96
50A 50
51A 51
52A 104 104A 104B
52B 52
54A 54
55A 110 110A
56A 112
56B 56 56C11
56C 56 56B11
57A 57
59A 59 59B2
59B 59 59A2
60A 360
60B 240
60C 120
60D 120
60E 60
60F 60
62A 62 62B3
62B 62 62A3
66A 132
66B 66
68A 68
69A 69 69B5
69B 69 69A5
70A 140
70B 70
71A 71 71B7
71B 71 71A7
78A 78
78B 78 78C7
78C 78 78B7
84A 84
84B 84
84C 84
87A 87 87B5
87B 87 87A5
88A 88 88B7
88B 88 88A7
92A 92 92B5
92B 92 92A5
93A 93 93B11
93B 93 93A11
94A 94 94B5
94B 94 94A5
95A 95 95B7
95B 95 95A7
104A 104 104B11
104B 104 104A11
105A 105
110A 110
119A 119 119B11
119B 119 119A11

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