Symplectic group S8(2)


Order = 47 377 612 800 = 216.35.52.7.17.
Mult = 1.
Out = 1.

Robert Wilson's ATLAS page for S8(2) is available here.

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

< x, y | x2 = y9 = (xy)17 = [x, y]2 = [x, y2]2 = [x, y4]3 = [x, y3xy3]2 = 1 >

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:

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.


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- Last updated 20th June, 1997
- John N. Bray