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
Th
Download

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

Representations

Representations of Th

Maximal subgroups

Maximal subgroups of Th

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

Conjugacy classes

Conjugacy classes of Th

Conjugacy class Centraliser order Power up Class rep(s)
1A90 745 943 887 872 000
2A92 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
3A12 737 088 6B 12A 12B 21A 24A 24B 39A 39B b
3B472 392 6C 9A 9B 9C 12C 18A 18B 27A 27B 27C 36A 36B 36C (ababab2)6
[a, bab]3
3C174 960 6A 12D 15A 15B 24C 24D 30A 30B [a, babab]2
4A387 072 8A 12A 12B 12C 24A 24B 28A 36A 36B 36C ababab2ababab2abab2
4B7 680 8B 12D 20A 24C 24D (abababab2ab2)3
5A3 000 10A 15A 15B 20A 30A 30B [a, b]2
(abab2)2
6A2 160 12D 24C 24D 30A 30B (ab)6(ab2)3
6B1 728 12A 12B 24A 24B (ab)6(ab2)3
6C648 12C 18A 18B 36A 36B 36C (ababab2)3
7A1 176 14A 21A 28A (ab)6(ab2)6
(abababab2)3
8A384 24A 24B (ab)7ab2
((ab)3ab2ab(ab2)2)3
8B96 24C 24D ababababab2(abab2ab2ab2)2
(ab)9(ab2)2ab(ab2)3
((ab)7ab2ab(ab2)2)3
9A5 832 18A 36A 36B 36C abab(abab2)3ab2ab2
9B729 27A 27B 27C ababababab2ab2abab2ab2
((ab)5ab2)3
9C162 18B (ababab2)2
[a, bab]
10A120 20A 30A 30B [a, b]
abab2
12A288 12B5 24A 24B
12B288 12A5 24A 24B
12C108 36A 36B 36C (abababab2ab2)2ab2
12D24 24C 24D abababab2ab2
13A39 39A 39B (ab)9(ab2)3
ab(abababab2)2
14A56 28A (ab)4(ab2)3
15A30 15B7 30A 30B
15B30 15A7 30A 30B
18A72 36A 36B 36C (ab)10(ab2)4
abababab2abab2ab2abab2
(ab)5(ababab2)2
18B18 abababb
19A19 ab
20A20 abababababb
21A21 ababababb
24A24 24B5
24B24 24A5
24C24 24D13
24D24 24C13
27A27 ababababababb
27B27 27C2
27C27 27B2
28A28 abababababababb
30A30 30B7
30B30 30A7
31A31 31B3
31B31 31A3
36A36 abababababbababababababababb
36B36 36C5
36C36 36B5
39A39 39B7
39B39 39A7
24A-B aabababbabababababababbabbabababababbababababababababb
24C-D abbabbabababababbababababababababb
27B-C abababbabababababababbabbabababababbababababababababb
30A-B babababababababbabbabababababbababababababababb
31A-B abababababababbabbabababababbababababababababb
36B-C abbabbabababababbababababababababbabababababbababababababababb
39A-B ababababbabababababababbabbabababababbababababababababb

Download words for class representatives.