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 The representations of 2.A13 available are The representations of S13 = A13:2 available are The representations of 2.S13 (plus type) available are The representations of 2.S13 (minus type) available are
Main ATLAS page Go to main ATLAS (version 2.0) page.
Alternating groups page Go to alternating groups page.
Old A13 page Go to old A13 page - ATLAS version 1.
ftp access 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.