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
At present three sets of generators of E_{6}(2) are in use: the set (a, b)
is labelled G1, the set (e, f) is labelled G2, and the set (x, y) is labelled G0 below.
Currently, we do not know conditions that would turn G0generators into a pair
of standard generators of E_{6}(2).
The G1standard generators of E_{6}(2) are a and b where a is
in class 2C, b is in class 3C, ab has order 62, ababb has order 24,
abababb has order 21 and abababab^{2}abab^{2}ab^{2} has order 18.
The G2standard generators of E_{6}(2) are e and f where e is
in class 2B, f is in class 3C, ef has order 62, efeff has order 12 and
efefeff has order 93.
Standard generators of E_{6}(2):2 are not defined.
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.
We may obtain (a conjugate in E_{6}(2):2 of) (e, f) by setting e = ([a, babab]^{10})^((ab)^{3}(ba)^{21}) and f = b.
We may obtain (a conjugate in E_{6}(2):2 of) (a, b) by setting a = ((efefefef^{2}efef^{2})^{21})^((fe)^{8}(ef)^{33}(fe)^{43}) and b = f
(which is not the inverse of the above program, as their composition is an inner element in class 12F).
An outer automorphism, u say, of E_{6}(2) maps (e, f) to (e, f^{−1}).
Potential standard generators are c and d where c = u is the above automorphism and
d = ((fef)^{3}e)^{17}.
These satisfy: c is in class 2E, d is in class 3B,
cd has order 16, cdcdd has order 31, cdcdcdd has order 10, cdcdcdcdd has order 73,
cdcdcddcdd of order 4, and so on, but we don't know if these conditions suffice.
Other potential standard generators are c = ((ef)^{6}effeffefu)^{13} and d = (ffefefefeffefefe)^{6}.
These satisfy: c is in class 2D, d has order 5 (class 5A),
cd has order 34, cdd has order 20, cdcdd has order 20, cdcddd has order 20, cdcdddd has order 3, cddcddd has order 7, cdcddcddd has order 40, cdcdddcdd has order 30,
and so on, but we don't know if these conditions suffice.
It is probably better to try to define standard generators on the latter pair of generators, or a similar pair to them.
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),
a and b (Magma).

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

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

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

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

Dimension 78 over GF(2):
x and
y (Meataxe),
x and
y (Meataxe binary),
x and
y (GAP).
The representations of E_{6}(2):2 available are:
Dimension 54 over GF(2):
e, f and u (Magma).
Dimension 78 over GF(2):
e, f and u (Magma).
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), 266–294.
 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}.
 (7 × ^{3}D_{4}(2)):3.
 G_{2}(2) — two classes.
Go to main ATLAS (version 2.0) page.
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Go to old E6(2) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 created on 21st April 1999.
Last updated 18.11.08 by JNB.
Information checked to
Level 0 on 21.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.