ATLAS: Alternating group A14

Order = 43589145600 = 210.35.52.72.11.13.
Mult = 2.
Out = 2.

The following information is available for A14:


Standard generators

Standard generators of A14 are a and b where a is in class 3A, b has order 13, ab has order 12, abb has order 24 and ababb has order 20. The last two conditions may be replaced by [a, b] has order 2.
In the natural representation we may take a = (1, 2, 3) and b = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14).
Standard generators of the double cover 2.A14 are preimages A and B where A has order 3 and B has order 13.

Standard generators of S14 = A14:2 are c and d where c is in class 2D, d has order 13 and ab has order 14.
In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14).
Standard generators of either of the double covers 2.S14 are preimages C and D where D has order 13.

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


Automorphisms

An outer automorphism of A14 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.

Black box algorithms

To find standard generators for A14: To find standard generators for S14 = A14.2:

Presentations

Presentations for A14 and S14 (respectively) on their standard generators are given below.

< a, b | a3 = b13 = (ab)12 = [a, b]2 = (aabab)2 = [a, babab]2 = (aabababab)2 = [a, babababab]2 = 1 >.

< c, d | c2 = d13 = (cd)14 = [c, d]3 = [c, dcd]2 = [c, dcdcd]2 = [c, (cd)4]2 = [c, (cd)5]2 = 1 >.

These presentations, and those of the covering groups, are available in Magma format as follows:
A14 on a and b, 2A14 on A and B, S14 on c and d, 2S14 (+) on C and D and 2S14 (-) on C and D.


Representations

Representations are available for the following decorations of A14. The representations of A14 available are: The representations of 2.A14 available are: The representations of S14 = A14:2 available are: The representations of 2.S14 (plus type) available are: The representations of 2.S14 (minus type) available are:

Maximal subgroups

The maximal subgroups of A14 are as follows: The maximal subgroups of S14 are as follows:

Conjugacy classes

[Some of] The 72 conjuagcy classes of A14 are as follows:

[Some of] The 135 conjuagcy classes of S14 are as follows:


Main ATLAS page Go to main ATLAS (version 2.0) page.
Alternating groups page Go to alternating groups page.
Old A14 page Go to old A14 page - ATLAS version 1.
ftp access Anonymous ftp access is also available on sylow.mat.bham.ac.uk.

Version 2.0 created on 7th May 1999.
Last updated 19.05.99 by JNB.
Information checked to Level 0 on 19.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.