# ATLAS: Alternating group A9

Order = 181440 = 26.34.5.7
Mult = 2.
Out = 2.

### Standard generators

Standard generators of A9 are a and b where a is in class 3A, b has order 7 and ab has order 9.
In the natural representation we may take a = (1, 2, 3) and b = (3, 4, 5, 6, 7, 8, 9).
Standard generators of the double cover 2.A9 are preimages A and B where A has order 3, and B has order 7.

### Representations

The representations of A9 available are
• All primitive permutation representations.
• a and b as the above permutations on 9 points - the natural representation.
• a and b as permutations on 36 points.
• a and b as permutations on 84 points.
• a and b as permutations on 120 points.
• a and b as permutations on 120 points.
• a and b as permutations on 126 points.
• a and b as permutations on 280 points.
• a and b as permutations on 840 points.
• Faithful irreducibles in characteristic 2.
• a and b as 8 × 8 matrices over GF(2) - the deleted permutation representation.
• a and b as 8 × 8 matrices over GF(2).
• a and b as 8 × 8 matrices over GF(2).
• a and b as 7 × 7 matrices over GF(3).
• a and b as 8 × 8 matrices over GF(5).
• a and b as 8 × 8 matrices over GF(7).
• a and b as 8 × 8 matrices over Z.
The representations of 2.A9 available are
• Permutation representations.
• A and B as permutations on 240 points.
• All faithful irreducibles in characteristic 3.
• A and B as 8 × 8 matrices over GF(3).
• A and B as 48 × 48 matrices over GF(3).
• A and B as 104 × 104 matrices over GF(3).
• Faithful irreducibles in characteristic 5.
• A and B as 8 × 8 matrices over GF(5).
• A and B as 8 × 8 matrices over GF(5).
• Faithful irreducibles in characteristic 7.
• A and B as 8 × 8 matrices over GF(7).
• A and B as 8 × 8 matrices over GF(7).
• Faithful irreducibles in characteristic 0.
• A and B as 8 × 8 matrices over Z.
• A and B as 8 × 8 matrices over Z.
The representations of S9 = A9:2 available are
• c and d as permutations on 9 points - the natural representation.