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)

Representations of 2.Sz(8)

Representations of Sz(8):3

Representations of 22.Sz(8):3


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)
1A29 120
2A64
4A16 b
4B16
5A5 ab
7A7 abb
7B7
7C7
13A13 ababbb
13B13
13C13

Download words for class representatives.

Conjugacy classes of Sz(8):3

Conjugacy class Centraliser order Power up Class rep(s)
1A87 360
2A192
4A48
4B48
5A15
7A7 cdcdcddcdcddcdd
13A13 cdcdd
3A60
3B60
6A12
6B12
12A12 cdcdcddcdcdd
12B12
12C12
12D12
15A15 cd
15B15

Download words for class representatives.