Order = 448345497600 = 213.37.52.7.11.13.
Mult = 6.
Out = 2.

## Porting notes

Porting incomplete.

## Standard generators

Standard generators of Suz are a, b where a is in class 2B, b is in class 3B, ab has order 13 and ababb has order 15.

Standard generators of 2.Suz are preimages A, B where B has order 3 and AB has order 13.

Standard generators of 3.Suz are preimages A, B where A has order 2 and AB has order 13.

Standard generators of 6.Suz are preimages A, B where A has order 4, B has order 3 and AB has order 13.

Standard generators of Suz:2 are c, d where c is in class 2C, d is in class 3B and cd has order 28.

Standard generators of 2.Suz:2 are preimages C, D where D has order 3.

Standard generators of 3.Suz:2 are preimages C, D where D is in class +3B. Alternatively: CDCDD has order 7.

Standard generators of 6.Suz:2 are preimages C, D where D has order 3 and CDCDD has order 7.

Standard generators of 6.Suz:2 (type b) are preimages C, D where D has order 3 and CDCDD has order 14.

## Automorphisms

Group Automorphism Order in Out(G) Description
Suz Outer automorphism u 2 a maps to aabab
b maps to babbabb
(No program)
Suz Outer automorphism v 2 a maps to a
b maps to bababbab
(No program)

(Suz) If c' = u15 and d'=b then (c',d') is conjugate to (c,d).

(Suz) v is in class 8H and ((babv)7, baba) is conjugate to (c,d).

## Black box algorithms

### Finding generators

Group Algorithm File

### Checking generators (semi-presentations)

Group Semi-presentation File
Suz 〈〈 a, b | o(a) = 2, o(b) = 3, o(ab) = 13, o(ababb) = 15, o(abababb) = 12 〉〉 Download
Suz:2 〈〈 c, d | o(c) = 2, o(d) = 3, o(cd) = 28, o(cdcdcddcdd) = 7 〉〉 Download

## Representations

### Representations of Suz

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
1782 Std Details
22880 Std Details
32760 Std Details
135135 Std Details
232960 Std Details
370656 Std Details
405405 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
0 Z 143 Std Details
Char Ring Dimension ID Generators Description Link
2 GF(4) 110 a Std Details
2 GF(2) 142 Std Details
2 GF(4) 572 a Std Details
2 GF(4) 572 b Std Details
2 GF(2) 638 Std Details
Char Ring Dimension ID Generators Description Link
3 GF(3) 64 Std Details
3 GF(3) 78 Std Details
3 GF(3) 286 Std Details
3 GF(3) 429 Std Details
3 GF(3) 649 Std Details
Char Ring Dimension ID Generators Description Link
5 GF(5) 143 Std Details
5 GF(5) 363 Std Details
5 GF(5) 780 Std Details
Char Ring Dimension ID Generators Description Link
7 GF(7) 143 Std Details
7 GF(7) 364 Std Details
7 GF(7) 780 Std Details
Char Ring Dimension ID Generators Description Link
11 GF(11) 143 Std Details
11 GF(11) 364 Std Details
11 GF(11) 779 Std Details
Char Ring Dimension ID Generators Description Link
13 GF(13) 143 Std Details
13 GF(13) 364 Std Details
13 GF(13) 780 Std Details

### Representations of 2.Suz

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
65520 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
3 GF(3) 12 Std Details
3 GF(3) 208 Std Details
3 GF(3) 352 Std Details

### Representations of 6.Suz

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
196560 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
5 GF(25) 12 a Std Details
7 GF(7) 12 a Std Details
13 GF(13) 12 a Std Details

### Representations of Suz:2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
1782 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
3 GF(3) 64 a Std Details
5 GF(5) 143 a Std Details
7 GF(7) 143 a Std Details
11 GF(11) 143 a Std Details

### Representations of 3.Suz:2

• View detailed report.
• Permutation representations:
Number of points ID Generators Description Link
5346 Std Details
• Matrix representations
Char Ring Dimension ID Generators Description Link
2 GF(2) 24 Std Details
5 GF(5) 132 Std Details
7 GF(7) 132 Std Details
11 GF(11) 132 Std Details

## Maximal subgroups

### Maximal subgroups of Suz

Subgroup Order Index Programs/reps
G2(4) 251 596 800 1 782 Program: Standard generators
32.U4(3).23' 19 595 520 22 880 Program: Generators
U5(2) 13 685 760 32 760 Program: Standard generators
21+6.U4(2) 3 317 760 135 135 Program: Generators
35:M11 1 924 560 232 960 Program: Generators mapping onto standard generators
J2:2 1 209 600 370 656 Program: Generators
24+6:3A6 1 105 920 405 405 Program: Generators
(A4 × L3(4)):2 483 840 926 640 Program: Generators
22+8:(A5 × S3) 368 640 1 216 215 Program: Generators
M12:2 190 080 2 358 720 Program: Standard generators
32+4:2.(A4 × 22).2 139 968 3 203 200 Program: Generators
(A6 × A5).2 43 200 10 378 368 Program: Generators
(A6 × 32:4).2 25 920 17 297 280 Program: Generators
L3(3):2 11 232 39 916 800 Program: Standard generators
L3(3):2 11 232 39 916 800 Program: Standard generators
L2(25) 7 800 57 480 192 Program: Generators
A7 2 520 177 914 880 Program: Generators

### Maximal subgroups of Suz:2

Subgroup Order Index Programs/reps
Suz 448 345 497 600 2 Program: Standard generators
G2(4):2 503 193 600 1 782 Program: Generators
Program: Generators
32.U4(3).22 39 191 040 22 880 Program: Generators
U5(2):2 27 371 520 32 760 Program: Standard generators
21+6.U4(2).2 6 635 520 135 135 Program: Generators mapping onto standard generators
Program: Generators
35:(M11 × 2) 3 849 120 232 960 Program: Generators mapping onto standard generators
J2:2 × 2 2 419 200 370 656 Program: Generators mapping onto standard generators
Program: Generators
24+6:3.S6 2 211 840 405 405 Program: Generators mapping onto standard generators
Program: Generators
(A4 × L3(4):2):2 967 680 926 640 Program: Generators
22+8:(S5 × S3) 737 280 1 216 215 Program: Generators
M12:2 × 2 380 160 2 358 720 Program: Generators mapping onto standard generators
Program: Generators
32+4:2.(S4 × D8) 279 936 3 203 200 Program: Generators
(PGL2(9) × A5).2 86 400 10 378 368 Program: Generators mapping onto standard generators
Program: Generators
(A6 × 32:8).2 51 840 17 297 280 Program: Generators mapping onto standard generators
L2(25):2 15 600 57 480 192 Program: Generators
S7 5 040 177 914 880 Program: Standard generators
Program: Generators

## Conjugacy classes

### Conjugacy classes of Suz

Conjugacy class Centraliser order Power up Class rep(s)
1A 448 345 497 600
2A 3 317 760 4A 4B 4C 6A 6B 6C 6D 8A 8B 8C 10A 12A 12B 12C 12E 18A 18B 20A 24A
2B 161 280 4D 6E 10B 12D 14A
3A 9 797 760 6A 12A 12C 15C 21A 21B 24A
3B 34 992 6B 6C 6D 9A 9B 12B 12E 18A 18B
3C 3 240 6E 12D 15A 15B
4A 46 080 8A 12A 12B 20A 24A
4B 3 072 8B 12E
4C 1 536 8C 12C
4D 288 12D
5A 1 800 10A 15A 15B 20A
5B 300 10B 15C
6A 3 456 12A 12C 24A
6B 1 296 6C5 18A 18B
6C 1 296 6B5 18A 18B
6D 432 12B 12E
6E 72 12D
7A 84 14A 21A 21B
8A 192 24A
8B 64 abababbababbabb
8C 32 abababbababb
9A 54 9B2 18A 18B
9B 54 9A2 18A 18B
10A 40 20A
10B 20 abababababbababbabb
11A 11 abababababbababbabbababbababababbababbabb
12A 288 24A
12B 72 abababb
12C 48 abababbabababababbababbabb
12D 36 ababbababababbababbabb
12E 24 abababbabababbababbabb
13A 13 13B2
13B 13 13A2
14A 28 ababababbababbabb
15A 45 15B2
15B 45 15A2
15C 15 ababababababbababbabb
18A 18 18B5
18B 18 18A5
20A 20 abababbababbabbabababbabababababbababbabb
21A 21 21B2
21B 21 21A2
24A 24 ababababababbababbabbababababbabababbababb
13A-B ab
15A-B ababb
18A-B ababababbabababbababb
21A-B ababababb

### Conjugacy classes of Suz:2

Conjugacy class Centraliser order Power up Class rep(s)
1A 896 690 995 200
2A 6 635 520 4A 4B 4C 6A 6B 6C 8A 8B 8C 10A 12A 12B 12C 12E 18A 20A 24A 4E 8D 8E 8F 8G 8H 12F 12G 16A 24B 24C 24D 24E 24F 40A 40B
2B 322 560 4D 6D 10B 12D 14A 4F 12H 28A
3A 19 595 520 6A 12A 12C 15B 21A 24A 6E 12F 24B 24C 24F 30A
3B 69 984 6B 6C 9A 12B 12E 18A 6F 12G 24D 24E
3C 6 480 6D 12D 15A 6G 6H 12H
4A 92 160 8A 12A 12B 20A 24A 8D 8E 8F 8G 24B 24C 24D 24E 40A 40B
4B 6 144 8B 12E 16A
4C 3 072 8C 12C 8H 24F
4D 576 12D
5A 3 600 10A 15A 20A 10D 40A 40B
5B 600 10B 15B 10C 10E 30A
6A 6 912 12A 12C 24A 12F 24B 24C 24F
6B 1 296 18A
6C 864 12B 12E 12G 24D 24E
6D 144 12D 12H
7A 168 14A 21A 14B 28A
8A 384 24A
8B 128 16A
8C 64
9A 54 18A
10A 80 20A 40A 40B
10B 40
11A 22 22A
12A 576 24A 24B 24C
12B 144 24D 24E
12C 96 24F
12D 72
12E 48
13A 13
14A 56 28A
15A 45
15B 30 30A
18A 18
20A 40 40A 40B
21A 21
24A 48
2C 2 419 200 6E 6G 10C 10D 14B 30A
2D 380 160 6F 6H 10E 22A
4E 4 608 12F 12G
4F 1 344 12H 28A
6E 4 320 30A
6F 216
6G 144
6H 144
8D 46 080 24B 24D 40A 40B
8E 3 072 24C
8F 768 24E
8G 256
8H 192 24F
10C 600 30A
10D 100
10E 40
12F 288
12G 72
12H 24
14B 28
16A 16
22A 22
24B 288
24C 96
24D 72
24E 24
24F 24
28A 28
30A 30
40A 40 40B7
40B 40 40A7