# ATLAS: Alternating group A10

Order = 1814400 = 27.34.52.7.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of A10 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.A10 are preimages A and B where A has order 3 and B has order 9.

Standard generators of S10 = A10.2 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.S10 are preimages C and D where D has order 9.

In the natural representations given here, we have a = cd-1cd = [c, d] and b = d.

### Automorphisms

An outer automorphism of A10 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 A10 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.A10 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 S10 = A10:2 available are:
• c and d as the above permutations on 10 points.