ATLAS: Alternating group A_{10}
Order = 1814400 = 2^{7}.3^{4}.5^{2}.7.
Mult = 2.
Out = 2.
Standard generators
Standard generators of A_{10} are a
and b where
a is in class 3A, b has order 9,
ab has order 8 and abb has order 12.
In the natural representation we may take
a = (1, 2, 3) and
b = (2, 3, 4, 5, 6, 7, 8, 9, 10).
Standard generators of the double cover 2.A_{10} are preimages A and B where
A has order 3 and B has order 9.
Standard generators of S_{10} = A_{10}.2 are c
and d where
c is in class 2C,
d has order 9
and cd has order 10.
In the natural representation we may take
c = (1, 2) and
d = (2, 3, 4, 5, 6, 7, 8, 9, 10).
Standard generators of either of the double covers 2.S_{10} are preimages C and D where
D has order 9.
In the natural representations given here, we have a = cd^{-1}cd = [c, d] and b = d.
Automorphisms
An outer automorphism of A_{10} 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_{10} available are:
- All primitive permutation representations.
- a and
b as
the above permutations on 10 points.
- a and
b as
permutations on 45 points.
- a and
b as
permutations on 120 points.
- a and
b as
permutations on 126 points.
- a and
b as
permutations on 210 points.
- a and
b as
permutations on 945 points.
- a and
b as
permutations on 2520 points.
- a and
b as
16 × 16 matrices over GF(2).
The representations of 2.A_{10} available are:
- A and
B as
16 × 16 matrices over GF(3).
- A and
B as
8 × 8 matrices over GF(5).
- A and
B as
56 × 56 matrices over GF(5).
- A and
B as
16 × 16 matrices over GF(7).
- A and
B as 16 × 16 matrices over Z.
The representations of S_{10} = A_{10}:2 available are:
- c and
d as
the above permutations on 10 points.
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Last updated 7th March 1998,
R.A.Wilson, R.A.Parker and J.N.Bray