ATLAS: Alternating group A_{8}, Linear group L_{4}(2)
Order = 20160 = 2^{6}.3^{2}.5.7.
Mult = 2.
Out = 2.
See also ATLAS of Finite Groups, p22.
Standard generators
Standard generators of A_{8} are a and b where
a is in class 3A, b has order 7, ab has order 6 and abb has order 15.
In the natural representation we may take
a = (1, 2, 3) and
b = (2, 3, 4, 5, 6, 7, 8).
Standard generators of the double cover 2.A_{8} are preimages A
and B where A has order 3 and B has order 7.
Standard generators of S_{8} = A_{8}:2 are c
and d where c is in class 2C, d has order 7
and cd has order 8.
In the natural representation, we may take
c = (1, 2) and
d = (2, 3, 4, 5, 6, 7, 8).
Standard generators of either of the double covers 2.S_{8} are
preimages C and D where D has order 7.
Representations
The representations of A_{8} = L_{4}(2) available are:

Permutations on 8 points  the natural representation above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 15 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 15 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducible representations in characteristic 2.

Dimension 4 over GF(2)  illustrating the isomorphism A8 = L4(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 4 over GF(2)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 6 over GF(2)  illustrating the isomorphism A8 = O6+(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(2)  the dual of the above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(2)  the Steinberg representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.A_{8} available are:

Permutations on 240 points  on the cosets of L2(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 240 points  on the cosets of 2^3:7:3:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 240 points  on the cosets of the other 2^3:7:3:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 3.

Dimension 8 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(9)  the dual of the above:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 48 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 48 over GF(3)  reducible over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 5.

Dimension 8 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 24 over GF(25)  the dual of the above:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(25)  the dual of the above:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 48 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 48 over GF(5)  reducible over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 64 over GF(5)  reducible over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 7.

Dimension 8 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 16 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 48 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(7)  the dual of the above:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 56 over GF(49)  the dual of the above:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 112 over GF(7)  reducible over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 Dimension 4 over Z_{4}:
A and B (Magma).
 Some faithful irreducibles in characteristic 0
The representations of S_{8} = A_{8}:2 available are:

Permutations on 8 points  the natural representation above:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 2.

Dimension 6 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 8 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 40 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 64 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
Go to main ATLAS (version 2.0) page.
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Go to old A8 page  ATLAS version 1.
Anonymous ftp access is also available on
sylow.mat.bham.ac.uk.
Version 2.0 created on 14th April 1999.
Last updated 11.03.04 by SJN.
Information checked to
Level 1 on 14.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.