# ATLAS: Alternating group A12

Order = 239500800 = 29.35.52.7.11.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of A12 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.A12 are preimages A and B where A has order 3 and B has order 11.

Standard generators of S12 = A12:2 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.S12 are preimages C and D where D has order 11.

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

### Automorphisms

An outer automorphism of A12 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 A12 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.A12 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 S12 = A12:2 available are
• c and d as the above permutations on 12 points.

Return to main ATLAS page. Last updated 24.03.99

R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk