ATLAS: Alternating group A_{12}
Order = 239500800 = 2^{9}.3^{5}.5^{2}.7.11.
Mult = 2.
Out = 2.
Standard generators
Standard generators of A_{12} are a and
b where a is in class 3A,
b has order 11, ab has order 10 and
abb has order 35.
In the natural representation we may take
a = (1, 2, 3) and
b = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12).
Standard generators of the double cover 2.A_{12} are preimages A and B where
A has order 3 and B has order 11.
Standard generators of S_{12} = A_{12}:2 are c and
d where c is in class 2D,
d has order 11 and ab has order 12.
In the natural representation we may take
c = (1, 2) and
d = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12).
Standard generators of either of the double covers 2.S_{12} are preimages C and D where
D has order 11.
In the natural representations given here, we have a = cd^{-1}cd = [c, d] and b = d.
Automorphisms
An outer automorphism of A_{12} may be realised by mapping (a, b) to (a^{-1}, ba^{-1}). In the natural representations given here, this outer automorphism is conjugation by c.
Representations
The representations of A_{12} available are
- Some primitive permutation representations.
- a and
b as
the above permutations on 12 points.
- a and
b as
permutations on 66 points.
- a and
b as
permutations on 220 points.
- a and
b as
permutations on 462 points.
- a and
b as
permutations on 495 points.
- a and
b as
permutations on 792 points.
- a and
b as
permutations on 2520 points.
- Some representations in characteristic 2.
- a and
b as
10 × 10 matrices over GF(2).
- a and
b as
16 × 16 matrices over GF(4).
- a and
b as
16 × 16 matrices over GF(4) - the dual of the above.
- a and
b as
44 × 44 matrices over GF(2).
- a and
b as
100 × 100 matrices over GF(2).
- Some representations in characteristic 3.
- a and
b as
10 × 10 matrices over GF(3).
- a and
b as
45 × 45 matrices over GF(3).
- a and
b as
54 × 54 matrices over GF(3).
- Some representations in characteristic 5.
- a and
b as
11 × 11 matrices over GF(5).
- a and
b as
43 × 43 matrices over GF(5).
- a and
b as
55 × 55 matrices over GF(5).
- a and
b as
89 × 89 matrices over GF(5).
The representations of 2.A_{12} available are
- A and
B as
16 × 16 matrices over GF(3).
- A and
B as
16 × 16 matrices over GF(3) - the dual of the above.
- A and
B as
32 × 32 matrices over GF(5).
- A and
B as
32 × 32 matrices over GF(7).
- A and
B as
32 × 32 matrices over GF(11).
The representations of S_{12} = A_{12}:2 available are
- c and
d as
the above permutations on 12 points.
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Last updated 24.03.99
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk