# ATLAS: Alternating group A8, Linear group L4(2)

Order = 20160 = 26.32.5.7.
Mult = 2.
Out = 2.

### 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
• Dimension 8 over Z: A and B (GAP).
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. 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.