# 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:
• a and b as the above permutations on 8 points.
• a and b as permutations on 15 points.
• a and b as permutations on 15 points.
• All faithful irreducibles in characteristic 2.
• a and b as 4 × 4 matrices over GF(2) - illustrating the isomorphism A8 = L4(2).
• a and b as 4 × 4 matrices over GF(2) - the dual of the above.
• a and b as 6 × 6 matrices over GF(2) - illustrating the isomorphism A8 = O6+(2).
• a and b as 14 × 14 matrices over GF(2).
• a and b as 20 × 20 matrices over GF(2).
• a and b as 20 × 20 matrices over GF(2) - the dual of the above.
• a and b as 64 × 64 matrices over GF(2) - the Steinberg representation.
The representations of 2.A8 available are:
• A and B as permutations on 240 points - on the cosets of L2(7).
• A and B as permutations on 240 points - on the cosets of 2^3:7:3.
• A and B as permutations on 240 points - on the cosets of the other 2^3:7:3.
• All faithful irreducibles in characteristic 3.
• A and B as 8 × 8 matrices over GF(3).
• A and B as 24 × 24 matrices over GF(9).
• A and B as 24 × 24 matrices over GF(9) - the dual of the above.
• A and B as 48 × 48 matrices over GF(3).
• A and B as 48 × 48 matrices over GF(3) - reducible over GF(9).
• All faithful irreducibles in characteristic 5.
• A and B as 8 × 8 matrices over GF(5).
• A and B as 24 × 24 matrices over GF(25).
• A and B as 24 × 24 matrices over GF(25) - the dual of the above.
• A and B as 32 × 32 matrices over GF(25).
• A and B as 32 × 32 matrices over GF(25) - the dual of the above.
• A and B as 48 × 48 matrices over GF(5).
• A and B as 48 × 48 matrices over GF(5) - reducible over GF(25).
• A and B as 64 × 64 matrices over GF(5) - reducible over GF(25).
• All faithful irreducibles in characteristic 7.
• A and B as 8 × 8 matrices over GF(7).
• A and B as 16 × 16 matrices over GF(7).
• A and B as 48 × 48 matrices over GF(7).
• A and B as 56 × 56 matrices over GF(7).
• A and B as 56 × 56 matrices over GF(7) - the dual of the above.
• A and B as 56 × 56 matrices over GF(49).
• A and B as 56 × 56 matrices over GF(49) - the dual of the above.
• A and B as 112 × 112 matrices over GF(7) - reducible over GF(49).
• A and B as 4 × 4 matrices over the integers modulo 4.
The representations of S8 = A8:2 available are:
• c and d as permutations on 8 points.
• All faithful irreducibles in characteristic 2.
• c and d as 6 × 6 matrices over GF(2).
• c and d as 8 × 8 matrices over GF(2).
• c and d as 14 × 14 matrices over GF(2).
• c and d as 40 × 40 matrices over GF(2).
• c and d as 64 × 64 matrices over GF(2).