ATLAS: Alternating group A_{13}
Order = 3113510400 = 2^{9}.3^{5}.5^{2}.7.11.13.
Mult = 2.
Out = 2.
Standard generators of A_{13} 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.A_{13} are preimages
A and B where A has order 3 and B has order 11.
Standard generators of S_{13} 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.S_{13} are
preimages C and D where CD has order 13.
An outer automorphism of A_{13} 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^{1}cd = [c, d] and b = dcd^{1}cdc.
Presentations of A_{13} and S_{13} on their standard generators are given below.
< a, b  a^{3} = a^{11} = (ab)^{13} = (aa^{b})^{2} = (ab^{2}ab^{2})^{2} = (ab^{3}ab^{3})^{2} = (ab^{4}ab^{4})^{2} = (ab^{5}ab^{5})^{2} = 1 >.
< c, d  c^{2} = d^{12} = (cd)^{13} = [c, d]^{3} = [c, dcd]^{2} = [c, (cd)^{3}]^{2} = [c, (cd)^{4}]^{2} = [c, (cd)^{5}]^{2} = 1 >.
The representations of A_{13} available are

Permutations on 13 points  the natural representation above:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 78 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 32 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 32 over GF(4)  the automorph of the above.
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 64 over GF(2)  reducible over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.A_{13} available are

Dimension 32 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(3)  the automorph of the above,
kindly provided by John Bray.
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(25)  the automorph of the above.
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 64 over GF(5)  reducible over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(49)  the automorph of the above.
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 64 over GF(7)  reducible over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(121)  the automorph of the above.
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 64 over GF(11)  reducible over GF(121):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 32 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of S_{13} = A_{13}:2 available are

Permutations on 13 points  the natural representation above:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 78 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2.S_{13} (plus type) available are
The representations of 2.S_{13} (minus type) available are
Go to main ATLAS (version 2.0) page.
Go to alternating groups page.
Go to old A13 page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 created on 2nd June 1999.
Last updated 15.01.02 by RAW.
Information checked to
Level 0 on 02.06.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.