# 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 pre-images 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 pre-images 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 pre-images E and F where EF has order 19.
Standard generators of 2.^2E6(2):2 are pre-images 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: