ATLAS: Exceptional group ^2E6(2)
Order = 76532479683774853939200.
Mult = 2 x 2 x 3.
Out = S3.
Standard generators and automorphisms
Standard generators of ^2E6(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.^2E6(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.^2E6(2) are preimages
A
and B where
A has order 2 and AB has order 19.
Standard generators of ^2E6(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.^2E6(2):2 are preimages
E
and F where
EF has order 19.
Standard generators of 2.^2E6(2):2 are preimages
E
and F where
EF has order 19.
An automorphism of order 3 may be obtained by mapping
(a,b) to
((ab)^2bab,(abb)^6b(abb)^6).
An automorphism of order 2 may be obtained by mapping
(a,b) to
(a,(abb)^9b(abb)^9).
Representations
^2E6(2) and covers
 The representation of ^2E6(2) available is

a
and
b as
78 x 78 matrices over GF(2).

The representation of 2.^2E6(2) available is

A
and
B as
1704 x 1704 matrices over GF(2).
The representation of 2^2.^2E6(2) available is

A
and
B as
1706 x 1706 matrices over GF(2).
 The representation of 3.^2E6(2) available is

A
and
B as
27 x 27 matrices over GF(4).
^2E6(2):2 and covers

The representation of ^2E6(2):2 available is

e and
f as
78 x 78 matrices over GF(2).

e and
f as
1938 x 1938 matrices over GF(3).

The representation of 3.^2E6(2):2 available is

E
and
F as
54 x 54 matrices over GF(2).

The representations of 2.^2E6(2):2 available are

E and
F as
1705 x 1705 matrices over GF(2).

E and
F as
2432 x 2432 matrices over GF(3).
^2E6(2):3 and covers
 The representation of ^2E6(2):3 available is

c
and
d as
78 x 78 matrices over GF(2).
 The representation of 3.^2E6(2).3 available is

C'
and
D' as
27 x 27 matrices over GF(4).
^2E6(2):S3 and covers

The representation of ^2E6(2):S3 available is

g
and
h as
78 x 78 matrices over GF(2).

The representation of 3.^2E6(2):S3 available is

G'
and
H' as
54 x 54 matrices over GF(2).
Maximal subgroups
The maximal subgroups of ^2E6(2) include:
The maximal subgroups of ^2E6(2):2 include:
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Last updated 01.10.98
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk