Order = 29120 = 26.5.7.13.
Mult = 22.
Out = 3.
Porting notes
Porting incomplete. Automorphisms need adding.Add notes about standard generators?
Standard generators
Standard generators of Sz(8) are a, b where a has order 2, b has order 4, ab has order 5, abb has order 7 and abab3ab2 has order 7.
Standard generators of 2.Sz(8) are preimages A, B where AB has order 5, ABB has order 7 and ABABBBABB has order 7.
Standard generators of 22.Sz(8) are preimages A, B where AB has order 5 and ABB has order 7.
Standard generators of Sz(8):3 are c, d where c has order 2, d has order 3, cd has order 15 and cdcdcdcddcdcddcdd has order 6.
Standard generators of 22.Sz(8):3 are preimages C, D where CDCDD has order 13.
Presentations
| Group | Presentation | Link |
|---|---|---|
| Sz(8) | 〈 a, b | a2 = b4 = (ab)5 = (abb)7 = (abab2ab2ab−1)5 = (abab−1ab2)7 = 1 〉 | Details |
| Sz(8) | 〈 a, b | a2 = b4 = (ab)5 = (ab2)7 = (abab−1ab2)7 = [a, b]13 = (abab2ab−1ab2ab2)5 = abab2ab2abab2[a,b]3(ab2ab−1ab2)2 = 1 〉 | Details |
| 2.Sz(8) | 〈 A, B | A2 = B4 = (AB)5 = (AB2)7 = (ABAB−1AB2)7 = 1 〉 | Details |
| Sz(8):3 | 〈 c, d | c2 = d3 = (cd)15 = [c, d]13 = (cdcdcd−1)6 = [c, dcd−1cd−1(cdcdcd−1)2] = [c, [c, dcdcdcd−1cd−1]] = (cd)6(cd−1)4(cd)4cd−1(cd)3(cd−1)5 = 1 〉 | Details |
| 22.Sz(8):3 | 〈 C, D | C2 = D3 = (CD)15 = [C, D]13 = (CDCDCD−1)6 = [C, [C, (DC)3D−1CD−1]] = (CD)5CD−1((CD)3CD−1CD)2(CDCD−1)3CD−1 = 1 〉 | Details |
Representations
Representations of Sz(8)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 65 Std Details 520 Std Details 560 Std Details 1456 Std Details 2080 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z[i] 14 a Std Details 0 Z[i] 14 b Std Details 0 Z 64 Std Details 0 C 65 a Std Details 0 C 65 b Std Details 0 C 65 c Std Details 0 Z 91 Std Details 0 Z[i] 105 Std Details Char Ring Dimension ID Generators Description Link 2 GF(8) 4 a Std Details 2 GF(8) 16 a Std Details 2 GF(2) 64 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 14 a Std Details 5 GF(5) 14 b Std Details 5 GF(5) 35 a Std Details 5 GF(5) 35 b Std Details 5 GF(5) 35 c Std Details 5 GF(5) 63 Std Details 5 GF(125) 65 a Std Details 5 GF(125) 65 b Std Details 5 GF(125) 65 c Std Details 5 GF(5) 195 Std Details Char Ring Dimension ID Generators Description Link 7 GF(49) 14 a Std Details 7 GF(7) 64 Std Details 7 GF(7) 91 Std Details 7 GF(7) 105 Std Details Char Ring Dimension ID Generators Description Link 13 GF(13) 14 a Std Details 13 GF(13) 14 b Std Details 13 GF(13) 35 Std Details 13 GF(13) 65 a Std Details 13 GF(13) 65 b Std Details 13 GF(13) 65 c Std Details 13 GF(13) 91 Std Details
Representations of 2.Sz(8)
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- Permutation representations:
Number of points ID Generators Description Link 1040 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 128 Std Details 5 GF(5) 8 Std Details 7 GF(7) 40 Std Details 13 GF(13) 16 Std Details 13 GF(13) 24 Std Details
Representations of Sz(8):3
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- Permutation representations:
Number of points ID Generators Description Link 65 Std Details 520 Std Details 560 Std Details 1456 Std Details 2080 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 2 GF(2) 12 Std Details 2 GF(2) 48 Std Details 2 GF(2) 64 Std Details 5 GF(5) 14 Std Details 5 GF(5) 63 Std Details 5 GF(5) 105 Std Details 5 GF(5) 195 Std Details 7 GF(49) 14 Std Details 13 GF(13) 14 Std Details
Representations of 22.Sz(8):3
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- Permutation representations:
Number of points ID Generators Description Link 2080 Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 5 GF(5) 24 Std Details 7 GF(7) 120 Std Details 13 GF(13) 48 Std Details
Maximal subgroups
Maximal subgroups of Sz(8)
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| 23+3:7 | Program: Generators | ||
| 13:4 | Program: Generators | ||
| 5:4 | Program: Generators | ||
| D14 | Program: Generators |
Maximal subgroups of Sz(8):3
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| Sz(8) | Program: Standard generators Program: Standard generators | ||
| 23+3:7:3 | Program: Generators | ||
| 13:12 = F156 | Program: Generators | ||
| 5:4 × 3 = F20 × 3 | Program: Generators | ||
| 7:6 = F42 | Program: Generators |
Conjugacy classes
Conjugacy classes of Sz(8)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 29 120 | ||
| 2A | 64 | ||
| 4A | 16 |
b | |
| 4B | 16 | ||
| 5A | 5 |
ab | |
| 7A | 7 |
abb | |
| 7B | 7 | ||
| 7C | 7 | ||
| 13A | 13 |
ababbb | |
| 13B | 13 | ||
| 13C | 13 |
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Conjugacy classes of Sz(8):3
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 87 360 | ||
| 2A | 192 | ||
| 4A | 48 | ||
| 4B | 48 | ||
| 5A | 15 | ||
| 7A | 7 |
cdcdcddcdcddcdd | |
| 13A | 13 |
cdcdd | |
| 3A | 60 | ||
| 3B | 60 | ||
| 6A | 12 | ||
| 6B | 12 | ||
| 12A | 12 |
cdcdcddcdcdd | |
| 12B | 12 | ||
| 12C | 12 | ||
| 12D | 12 | ||
| 15A | 15 |
cd | |
| 15B | 15 |
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