Order = 90745943887872000 = 215.310.53.72.13.19.31.
Mult = 1.
Out = 1.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of Th are a, b where a has order 2, b is in class 3A and ab has order 19.

## Black box algorithms

### Finding generators

Group Algorithm File

### Checking generators (semi-presentations)

Group Semi-presentation File
Th 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 19, o(z) = 21, o(b(z7)w) = 2; z = (ab)3b, w = abb(ab)4(abb)2(ab)2(abb)5(ab)3 〉〉 Download

## Maximal subgroups

### Maximal subgroups of Th

Subgroup Order Index Programs/reps
3D4(2):3 634 023 936 143 127 000 Program: Generators
25.L5(2) 319 979 520 283 599 225 Program: Standard generators
21+8.A9 92 897 280 976 841 775 Program: Generators
U3(8):6 33 094 656 2 742 012 000 Program: Standard generators
Program: Standard generators
(3 × G2(3)):2 25 474 176 3 562 272 000 Program: Generators mapping onto standard generators
3.32.3.(3 × 32).32:2S4 944 784 96 049 408 000 Program: Generators
32.33.32.32:2S4 944 784 96 049 408 000 Program: Generators
35:2.S6 349 920 259 333 401 600 Program: Generators
51+2:4S4 12 000 7 562 161 990 656 Program: Generators
52:GL2(5) 12 000 7 562 161 990 656 Program: Generators
72:(3 × 2S4) 7 056 12 860 819 712 000 Program: Generators
L2(19):2 6 840 13 266 950 860 800 Program: Generators
L3(3) 5 616 16 158 465 792 000 Program: Generators
M10 720 126 036 033 177 600 Program: Generators
Program: Standard generators
F465 = 31:15 465 195 152 567 500 800 Program: Generators
Program: Same elements of Th but much shorter program
S5 120 756 216 199 065 600 Program: Standard generators

## Conjugacy classes

### Conjugacy classes of Th

Conjugacy class Centraliser order Power up Class rep(s)
1A 90 745 943 887 872 000
2A 92 897 280 4A 4B 6A 6B 6C 8A 8B 10A 12A 12B 12C 12D 14A 18A 18B 20A 24A 24B 24C 24D 28A 30A 30B 36A 36B 36C a
3A 12 737 088 6B 12A 12B 21A 24A 24B 39A 39B b
3B 472 392 6C 9A 9B 9C 12C 18A 18B 27A 27B 27C 36A 36B 36C (ababab2)6
[a, bab]3
3C 174 960 6A 12D 15A 15B 24C 24D 30A 30B [a, babab]2
4A 387 072 8A 12A 12B 12C 24A 24B 28A 36A 36B 36C ababab2ababab2abab2
4B 7 680 8B 12D 20A 24C 24D (abababab2ab2)3
5A 3 000 10A 15A 15B 20A 30A 30B [a, b]2
(abab2)2
6A 2 160 12D 24C 24D 30A 30B (ab)6(ab2)3
6B 1 728 12A 12B 24A 24B (ab)6(ab2)3
6C 648 12C 18A 18B 36A 36B 36C (ababab2)3
7A 1 176 14A 21A 28A (ab)6(ab2)6
(abababab2)3
8A 384 24A 24B (ab)7ab2
((ab)3ab2ab(ab2)2)3
8B 96 24C 24D ababababab2(abab2ab2ab2)2
(ab)9(ab2)2ab(ab2)3
((ab)7ab2ab(ab2)2)3
9A 5 832 18A 36A 36B 36C abab(abab2)3ab2ab2
9B 729 27A 27B 27C ababababab2ab2abab2ab2
((ab)5ab2)3
9C 162 18B (ababab2)2
[a, bab]
10A 120 20A 30A 30B [a, b]
abab2
12A 288 12B5 24A 24B
12B 288 12A5 24A 24B
12C 108 36A 36B 36C (abababab2ab2)2ab2
12D 24 24C 24D abababab2ab2
13A 39 39A 39B (ab)9(ab2)3
ab(abababab2)2
14A 56 28A (ab)4(ab2)3
15A 30 15B7 30A 30B
15B 30 15A7 30A 30B
18A 72 36A 36B 36C (ab)10(ab2)4
abababab2abab2ab2abab2
(ab)5(ababab2)2
18B 18 abababb
19A 19 ab
20A 20 abababababb
21A 21 ababababb
24A 24 24B5
24B 24 24A5
24C 24 24D13
24D 24 24C13
27A 27 ababababababb
27B 27 27C2
27C 27 27B2
28A 28 abababababababb
30A 30 30B7
30B 30 30A7
31A 31 31B3
31B 31 31A3
36A 36 abababababbababababababababb
36B 36 36C5
36C 36 36B5
39A 39 39B7
39B 39 39A7
24A-B aabababbabababababababbabbabababababbababababababababb
24C-D abbabbabababababbababababababababb
27B-C abababbabababababababbabbabababababbababababababababb
30A-B babababababababbabbabababababbababababababababb
31A-B abababababababbabbabababababbababababababababb
36B-C abbabbabababababbababababababababbabababababbababababababababb
39A-B ababababbabababababababbabbabababababbababababababababb