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 |
|---|---|---|
| Suz |
| Download |
| Suz:2 |
| Download |
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 3.Suz
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 5346 Std Details 98280 Std Details 405405 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(4) 12 a Std Details 2 GF(4) 66 a Std Details 2 GF(4) 429 a Std Details 2 GF(4) 825 a Std Details 2 GF(4) 825 b Std Details 5 GF(25) 66 a 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 2.Suz:2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 12 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
Representations of 6.Suz:2
- View detailed report.
- Matrix representations
Char Ring Dimension ID Generators Description Link 3 GF(3) 24 Std Details 5 GF(5) 24 Std Details 11 GF(11) 24 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 |
Download words for class representatives.
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 |
Download words (if any exist) for class representatives.