ATLAS: Alternating group A8, Linear group L4(2)
Order = 20160 = 26.32.5.7.
Mult = 2.
Out = 2.
See also ATLAS of Finite Groups, p22.
Standard generators
Standard generators of A8 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.A8 are preimages A
and B where A has order 3 and B has order 7.
Standard generators of S8 = A8:2 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.S8 are
preimages C and D where D has order 7.
Representations
The representations of A8 = L4(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.A8 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 Z4:
A and B (Magma).
- Some faithful irreducibles in characteristic 0
The representations of S8 = A8:2 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.
Go to alternating groups page.
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.