# ATLAS: Alternating group A13

Order = 3113510400 = 29.35.52.7.11.13.
Mult = 2.
Out = 2.

### Standard generators

Standard generators of A13 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.A13 are preimages A and B where A has order 3 and B has order 11.

Standard generators of S13 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.S13 are preimages C and D where CD has order 13.

### Automorphisms

An outer automorphism of A13 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-1cd = [c, d] and b = dcd-1cdc.

### Presentations

Presentations of A13 and S13 on their standard generators are given below.

< a, b | a3 = a11 = (ab)13 = (aab)2 = (ab-2ab2)2 = (ab-3ab3)2 = (ab-4ab4)2 = (ab-5ab5)2 = 1 >.

< c, d | c2 = d12 = (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 A13 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.A13 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 S13 = A13:2 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.S13 (plus type) available are
• none.
The representations of 2.S13 (minus type) available are
• none.