Order = 4157776806543360000 = 221.39.54.72.11.13.23.
Mult = 2.
Out = 1.

Porting notes

Fully copied from version 2. 30/7/06.

Standard generators

Standard generators of Co1 are a, b where a is in class 2B, b is in class 3C, ab has order 40 and ababb has order 6.

Standard generators of 2.Co1 = Co0 are preimages A, B where B has order 3 and ABABABABBABABBAB has order 33.

Black box algorithms

Finding generators

Group Algorithm File
Co1 Download

Checking generators (semi-presentations)

Group Semi-presentation File
Co1 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 40, o(abab2) = 6, o(z) = 42, o(ab2z21) = 11, o(uv) = 36, o(uvvuvuv) = 18; z = ab(ababb)2, u = (ab)20, v = bababababb 〉〉 Download

Representations

Representations of Co1

Representations of 2.Co1

Maximal subgroups

Maximal subgroups of Co1

Subgroup Order Index Programs/reps
Co2 42 305 421 312 000 98 280 Program: Standard generators
3.Suz:2 2 690 072 985 600 1 545 600 Program: Standard generators
211:M24 501 397 585 920 8 292 375 Program: Generators
Co3 495 766 656 000 8 386 560 Program: Generators
21+8.O8+(2) 89 181 388 800 46 621 575 Program: Generators
U6(2):S3 55 180 984 320 75 348 000 Program: Generators
(A4 × G2(4)):2 6 038 323 200 688 564 800
22+12:(A8 × S3) 1 981 808 640 2 097 970 875
24+12.(S3 × 3.S6) 849 346 560 4 895 265 375
32.U4(3).D8 235 146 240 17 681 664 000
36:2.M12 138 568 320 30 005 248 000
(A5 × J2):2 72 576 000 57 288 591 360
31+4:2.S4(3).2 25 194 240 165 028 864 000
(A6 × U3(3)).2 4 354 560 954 809 856 000
33+4:2.(S4 × S4) 2 519 424 1 650 288 640 000
A9 × S3 1 088 640 3 819 239 424 000
(A7 × L2(7)):2 846 720 4 910 450 688 000
(D10 × (A5 × A5).2).2 144 000 28 873 450 045 440
51+2:GL2(5) 60 000 69 296 280 109 056
53:(4 × A5).2 60 000 69 296 280 109 056
72:(3 × 2.S4) 3 528 1 178 508 165 120 000
52:2A5 3 000 1 385 925 602 181 120

Conjugacy classes

Conjugacy classes of Co1

Conjugacy class Centraliser order Power up Class rep(s)
1A 4 157 776 806 543 360 000
2A 89 181 388 800 4A 4B 4C 4D 6A 6C 6D 6E 6F 8A 8B 8C 8D 8E 8F 10A 10D 10E 12A 12B 12C 12D 12E 12G 12H 12I 12J 12K 14B 16A 16B 18A 18B 18C 20A 20C 24A 24B 24C 24D 24E 24F 28A 30A 30D 30E 36A 40A 60A
2B 2 012 774 400 4E 6B 6H 10B 10C 12F 12L 14A 26A 28B 30B 30C 42A
2C 389 283 840 4F 6G 6I 10F 12M 20B 22A
3A 1 345 036 492 800 6A 6B 12A 12B 12C 12F 15A 15B 21A 24A 24B 24D 30A 30B 33A 39A 39B 42A 60A
3B 58 786 560 6C 6E 6G 12E 12G 12I 12J 15D 21B 24C 24E 30D
3C 12 597 120 6D 6F 9A 9B 9C 12D 12H 12K 15E 18A 18B 18C 24F 30E 36A
3D 544 320 6H 6I 12L 12M 15C 21C 30C
4A 371 589 120 8A 8B 12A 12D 12E 20A 24A 24C 24E 28A 36A 40A 60A
4B 1 179 648 8C 8D 12B 12G 12K 16A 16B 24B
4C 737 280 8E 12H 12I 20C 24F
4D 294 912 8F 12C 12J 24D
4E 48 384 12F 12L 28B
4F 15 360 12M 20B
5A 3 024 000 10A 10B 15A 15C 20A 30A 30C 35A 40A 60A
5B 36 000 10C 10D 10F 15B 15D 20B 30B 30D
5C 15 000 10E 15E 20C 30E
6A 9 953 280 12A 12B 12C 24A 24B 24D 30A 60A
6B 483 840 12F 30B 42A
6C 155 520 30D
6D 155 520 12H 24F 30E
6E 20 736 12E 12G 12I 12J 24C 24E
6F 15 552 12D 12K 18A 18B 18C 36A
6G 3 456 Omitted owing to length.
6H 1 440 12L 30C
6I 576 12M
7A 17 640 14A 21A 21B 28B 35A 42A
7B 1 176 14B 21C 28A
8A 92 160 24A 24C 40A
8B 6 144 24E
8C 6 144 16A 24B
8D 2 048 16B
8E 768 24F
8F 768 24D
9A 1 944 18A 36A
9B 486 18B
9C 486 18C
10A 9 600 20A 30A 40A 60A
10B 1 200 30C
10C 1 200 30B
10D 600 30D
10E 600 20C 30E
10F 160 20B
11A 66 22A 33A
12A 138 240 24A 60A
12B 9 216 24B
12C 4 608 24D
12D 2 592 36A
12E 1 728 24C 24E
12F 864 ababababbababbabababbabababbabababbababbabababbabababb
12G 576 ababbabababbababbabababbababbababababbababbabababb
12H 288 24F
12I 288 ababbabababbababbababababbababbabababb
12J 288 abababbabababbabababbabababbababbabababb
12K 288 Omitted owing to length.
12L 72 abaababbabababbabababbabababbababbabababb
12M 48 abababb
13A 156 26A 39A 39B
14A 168 28B 42A
14B 56 28A
15A 5 400 30A 60A
15B 900 30B
15C 180 30C
15D 90 30D
15E 30 30E
16A 64 Omitted owing to length.
16B 64 abababbababbababbabababbabababbabababbababbabababb
18A 216 36A
18B 54 ababababbabababbababbabababb
18C 54 Omitted owing to length.
20A 480 40A 60A
20B 40 Omitted owing to length.
20C 20 bababbabababbabababbabababbababbabababb
21A 252 42A
21B 63 aabbabababbabababbababbabababb
21C 21 abababbababbabababb
22A 22 ababbabababbabababbababbabababb
23A 23 23B5
23B 23 23A5
24A 576 ababbababababbababbabababbababbabababbababababbababbabababb
24B 192 Omitted owing to length.
24C 144 Omitted owing to length.
24D 96 ababababbababbabababb
24E 48 ababbabababbababababbabababbababbabababb
24F 24 abbabababbabababbababbabababb
26A 52 abababbababbabababbabababbabababbabababbabababbababbabababb
28A 28 Omitted owing to length.
28B 28 abababbabababbababbabababb
30A 120 60A
30B 60 ababbababababbababbabababb
30C 60 aababbabababbabababbabababbababbabababb
30D 30 ababbabababbababababbababbabababb
30E 30 ababababbababbabababbabbabababbabababbababbabababb
33A 33 abbababbabababbabababbabababbababbabababb
35A 35 ababbabababbabababbabababbababbabababbababababbababbabababb
36A 36 Omitted owing to length.
39A 39 39B7
39B 39 39A7
40A 40 ab
42A 84 ababbabababb
60A 60 abababbababbabababbababababbababbabababb
23A-B ababbabbababbabababbabababbabababbababbabababb
39A-B ababbababbabababbabababbabababbababbabababb

Download words for class representatives.