Order = 29120 = 26.5.7.13.
Mult = 22.
Out = 3.

## 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

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 22.Sz(8):3

• View detailed report.
• 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

### 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