Symplectic group S₈(2)

S₈(2): Length 127, 2-generator, 7-relator.

Remark: These are (2A, 9B, 17)-generators. We have that xy² has order 15. (It may be quite a challenge to find a presentation in terms of standard generators.) The presentation is available in MAGMA code here, where we have included the redundant relations [x, y³]³ = [x, yxy]² = 1 for ease of coset enumeration. (We have included the subgroups below also.)

Some subgroups:

• M1 = < x, yxy³ > = O8^-(2):2 of order 394 813 440 and index 120.
• M2 = < x, y³xy⁴ > = O8⁺(2):2 of order 348 364 800 and index 136.
• H1 = < yxy³, xyxy³x > = O8^-(2) of order 197 406 720 and index 240.
• H2 = < y³xy⁴, xy³xy⁴x > = O8⁺(2) of order 174 182 400 and index 272.

Realisation:
x =
1 0 0 0 0 0 0 0
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
1 0 0 1 0 0 0 0
1 0 0 0 1 0 0 0
1 0 0 0 0 1 0 0
1 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1

y =
0 1 0 0 0 0 0 0
0 0 1 0 0 0 0 0
0 0 0 1 0 0 0 0
0 0 0 0 1 0 0 0
0 0 0 0 0 1 0 0
0 0 0 0 0 0 1 0
0 0 0 0 0 0 0 1
1 1 1 1 1 1 1 1

Symplectic form: B =
0 0 0 1 1 1 1 0
0 0 0 0 1 1 1 1
0 0 0 0 0 1 1 1
1 0 0 0 0 0 1 1
1 1 0 0 0 0 0 1
1 1 1 0 0 0 0 0
1 1 1 1 0 0 0 0
0 1 1 1 1 0 0 0

These matrices are taken over the field F = GF(2).

The above matrices and form are available in MAGMA format here.

John N. Bray