ATLAS: Alternating group A_{13}
Order = 3113510400 = 2^{9}.3^{5}.5^{2}.7.11.13.
Mult = 2.
Out = 2.
Standard generators
Standard generators of A_{13} are a
and b where
a is in class 3A, b has order 11
and ab has order 13.
In the natural representation we may take
a = (1, 2, 3) and
b = (3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
Standard generators of the double cover 2.A_{13} are preimages
A and B where
A has order 3 and B has order 11.
Standard generators of S_{13} are c
and d where
c is in class 2D, d is in class 12M
and cd has order 13.
In the natural representation we may take
c = (1, 2) and
d = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13).
Standard generators of either of the double covers 2.S_{13} are
preimages C and D where
CD has order 13.
Automorphisms
An outer automorphism of A_{13} of order 2 may be obtained by mapping (a, b) to (a^{-1}, b).
In the natural representations given here, this outer automorphism is conjugation by c. We may then obtain d as d = bac.
Conversely, we have a = cd^{-1}cd = [c, d] and b = dcd^{-1}cdc.
Presentations
Presentations of A_{13} and S_{13} on their standard generators are given below.
< a, b | a^{3} = a^{11} = (ab)^{13} = (aa^{b})^{2} = (ab^{-2}ab^{2})^{2} = (ab^{-3}ab^{3})^{2} = (ab^{-4}ab^{4})^{2} = (ab^{-5}ab^{5})^{2} = 1 >.
< c, d | c^{2} = d^{12} = (cd)^{13} = [c, d]^{3} = [c, dcd]^{2} = [c, (cd)^{3}]^{2} = [c, (cd)^{4}]^{2} = [c, (cd)^{5}]^{2} = 1 >.
Representations
The representations of A_{13} available are
- a and
b as
the above permutations on 13 points - the natural representation.
- a and
b as
permutations on 78 points.
- a and
b as
32 × 32 matrices over GF(4).
- a and
b as
32 × 32 matrices over GF(4) - the automorph of the above.
- a and
b as
64 × 64 matrices over GF(2) - reducible over GF(4).
The representations of 2.A_{13} available are
- A and
B as
32 × 32 matrices over GF(3).
- A and
B as
32 × 32 matrices over GF(3) - kindly provided by J.N.Bray.
- A and
B as
32 × 32 matrices over GF(25).
- A and
B as
32 × 32 matrices over GF(25) - the automorph of the above.
- A and
B as
64 × 64 matrices over GF(5) - reducible over GF(25).
- A and
B as
32 × 32 matrices over GF(49).
- A and
B as
32 × 32 matrices over GF(49) - the automorph of the above.
- A and
B as
64 × 64 matrices over GF(7) - reducible over GF(49).
- A and
B as
32 × 32 matrices over GF(121).
- A and
B as
32 × 32 matrices over GF(121) - the automorph of the above.
- A and
B as
64 × 64 matrices over GF(11) - reducible over GF(121).
- A and
B as
32 × 32 matrices over GF(13).
The representations of S_{13} = A_{13}:2 available are
- c and
d as permutations on 13 points - the natural representation.
- c and
d as permutations on 78 points.
The representations of 2.S_{13} (plus type) available are
The representations of 2.S_{13} (minus type) available are
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Last updated 19.12.97
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk