ATLAS: Exceptional group ^{2}E_{6}(2)
Order = 76532479683774853939200 = 2^{36}.3^{9}.5^{2}.7^{2}.11.13.17.19.
Mult = 2^{2} × 3.
Out = S_{3}.
Standard generators of ^{2}E_{6}(2) are a and b
where a is in class 2B, b is in class 3C,
ab has order 19 and abababb has order 33.
Standard generators of the double cover 2.^{2}E_{6}(2) are
preimages A and B where B has order 3, AB has
order 19 and ABABABB has order 33.
Standard generators of the triple cover 3.^{2}E_{6}(2) are
preimages A and B where A has order 2 and AB
has order 19.
Standard generators of ^{2}E_{6}(2):2 are e and
f where e is in class 2D, f is in class 8R,
ef has order 19, and efeff has order 30.
Standard generators of 3.^{2}E_{6}(2):2 are preimages
E and F where EF has order 19.
Standard generators of 2.^{2}E_{6}(2):2 are preimages
E and F where EF has order 19.
Standard generators for the other groups below are not defined.
An automorphism of order 3 of ^{2}E_{6}(2) may be obtained by mapping
(a, b) to
((ab)^2bab, (abb)^6b(abb)^6).
An automorphism of order 2 of ^{2}E_{6}(2) may be obtained by mapping
(a, b) to
(a, (abb)^9b(abb)^9).
^{2}E_{6}(2) and covers
 The representations of ^{2}E_{6}(2) available are:

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

Dimension 1705 (structure 1702.(1+1+1)) over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 The representation of 2.^{2}E_{6}(2) available is:

Dimension 1704 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 The representation of 2^{2}.^{2}E_{6}(2) available is:

Dimension 1706 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 The representation of 3.^{2}E_{6}(2) available is:

Dimension 27 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
^{2}E_{6}(2):2 and covers
 The representation of ^{2}E_{6}(2):2 available is:

Dimension 78 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 1938 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
 The representation of 3.^{2}E_{6}(2):2 available is:

Dimension 54 over GF(2):
E and
F (Meataxe),
E and
F (Meataxe binary),
E and
F (GAP).
 The representations of 2.^{2}E_{6}(2):2 available are:

Dimension 1705 over GF(2):
E and
F (Meataxe),
E and
F (Meataxe binary),
E and
F (GAP).

Dimension 2432 over GF(3):
E and
F (Meataxe),
E and
F (Meataxe binary),
E and
F (GAP).
^{2}E_{6}(2):3 and covers
 The representation of ^{2}E_{6}(2):3 available is:

Dimension 78 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 The representation of 3.^{2}E_{6}(2).3 available is:

Dimension 27 over GF(4):
C' and
D' (Meataxe),
C' and
D' (Meataxe binary),
C' and
D' (GAP).
^{2}E_{6}(2):S_{3} and covers
 The representation of ^{2}E_{6}(2):S_{3} available is:

Dimension 78 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
 The representation of 3.^{2}E_{6}(2):S_{3} available is:

Dimension 54 over GF(2):
G' and
H' (Meataxe),
G' and
H' (Meataxe binary),
G' and
H' (GAP).
The maximal subgroups of ^{2}E_{6}(2) include:
The maximal subgroups of ^{2}E_{6}(2):2 include:
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Version 2.0 created on 26th April 2000.
Last updated 06.01.04 by RAW.
Information checked to
Level 0 on 23.05.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.