ATLAS: Alternating group A_{8}, Linear group L_{4}(2)
Order = 20160 = 2^{6}.3^{2}.5.7.
Mult = 2.
Out = 2.
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:
- 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.A_{8} 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 S_{8} = A_{8}: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).
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Last updated 14th April 1999,
R.A.Wilson, R.A.Parker and J.N.Bray