ATLAS: Symplectic group S8(2)

Order = 47377612800 = 216.35.52.7.17.
Mult = 1.
Out = 1.

Standard generators

Standard generators of S8(2) are a and b where a is in class 2B, b is in class 5B, ab has order 17 and abb has order 21.

Presentation

A presentation for S8(2) on its standard generators is given below.

< a, b | a2 = b5 = (ab)17 = [a, b]3 = [a, b2]4 = [a, bab]3 = [a, b2abab2]2 = [a, b2ab2]3 = [a, (ab2)4]2 = [a, babab]3 = (ababab2ab-1ab-2)4 = 1 >.


Representations

The representations of S8(2) available are:
Degree Type/Ring Generators Name Info Comments (if any)
Permutation representations
120 permutation a b S82p120 - 2-transitive (in fact, 2-primitive)
136 permutation a b S82p136 - 2-transitive (in fact, 2-primitive)
240 permutation a b S82p240 - imprimitive
255 permutation a b S82p255 - -
272 permutation a b S82p272 - imprimitive
2295 permutation a b S82p2295 - -
5355 permutation a b S82p5355 - -
Faithful irreducibles in characteristic 2
8 GF(2) a b S82f2r8 - The natural representation
16 GF(2) a b S82f2r16 - -
26 GF(2) a b S82f2r26 - -
48 GF(2) a b S82f2r48 - -
Faithful irreducibles in characteristic 3
35 GF(3) a b S82f3r35 - -
50 GF(3) a b S82f3r50 - -
Faithful irreducibles in characteristic 5
35 GF(5) a b S82f5r35 - -
51 GF(5) a b S82f5r51 - -
Faithful irreducibles in characteristic 7
35 GF(7) a b S82f7r35 - -
51 GF(7) a b S82f7r51 - -
Faithful irreducibles in characteristic 17
35 GF(17) a b S82f17r35 - -
51 GF(17) a b S82f17r51 - -
Faithful irreducible in characteristic 0
35 Z a and b S82Zr35 - -

Maximal subgroups


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Last updated 7th December 1998,
R.A.Wilson, R.A.Parker and J.N.Bray