ATLAS: Exceptional group E_{6}(2)
Order = 214841575522005575270400 =
2^{36}.3^{6}.5^{2}.7^{3}.13.17.31.73.
Mult = 1.
Out = 2.
Standard generators
Standard generators of E_{6}(2) and E_{6}(2):2 are not defined.
At present two sets of generators are in use: the set (a, b)
is labelled G1, and the set (x, y) is labelled G0 below.
We may obtain (a conjugate in E_{6}(2):2 of) (a, b) by setting a = ((xyxy^{2})^{6})^(y^{5}xy^{6}x^{2}y^{4}) and b = x.
Representations
The representations of E_{6}(2) available are:

Dimension 27 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(2)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 78 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(2):
x and
y (Meataxe),
x and
y (Meataxe binary),
x and
y (GAP).

Dimension 27 over GF(2)  the dual of the above:
x and
y (Meataxe),
x and
y (Meataxe binary),
x and
y (GAP).

Dimension 78 over GF(2):
x and
y (Meataxe),
x and
y (Meataxe binary),
x and
y (GAP).
Maximal subgroups
Taken from:
Peter Kleidman and Robert Wilson,
The maximal subgroups of E_{6}(2) and Aut(E_{6}(2)),
Proc London Math Soc 60 (1990), 266294.
 2^{16}:O_{10}^{+}(2)  two classes.
 2^{5+20}:(S_{3} × L_{5}(2))  two classes.
 2^{1+20}:L_{6}(2).
 [2^{29}]:(S_{3} × L_{3}(2) × L_{3}(2)).
 F_{4}(2).
 S_{3} × L_{6}(2).
 3.(3^{2}:Q_{8} × L_{3}(4)).S_{3}.
 L_{3}(8):3.
 (L_{3}(2) × L_{3}(2) × L_{3}(2)):S_{3}.
 L_{3}(2) × G_{2}(2).
 7^{3}:3^{1+2}:2A_{4}.
 G_{2}(2)  two classes.
Go to main ATLAS (version 2.0) page.
Go to exceptional groups page.
Go to old E6(2) page  ATLAS version 1.
Anonymous ftp access is also available on
sylow.mat.bham.ac.uk.
Version 2.0 created on 21st April 1999.
Last updated 17.05.00 by JNB.
Information checked to
Level 0 on 21.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.