ATLAS: Alternating group A7

Order = 2520 = 23.32.5.7.
Mult = 6.
Out = 2.
See also ATLAS of Finite Groups p10

The following information is available for A7:


Standard generators

Standard generators of A7 are a and b where a is in class 3A, b has order 5 and ab has order 7.
In the natural representation we may take a = (1, 2, 3) and b = (3, 4, 5, 6, 7).
Standard generators of the double cover 2.A7 are preimages A and B where A has order 3, B has order 5 and AB has order 7. Any two of these conditions implies the third.
Standard generators of the triple cover 3.A7 are preimages A and B where B has order 5 and AB has order 7.
Standard generators of the sextuple cover 6.A7 are preimages A and B where B has order 5 and AB has order 7.

Standard generators of S7 are c and d where c is in class 2B, d is in class 6C and cd has order 7.
In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5, 6, 7).
Standard generators of either of the double covers 2.S7 are preimages C and D where CD has order 7.
Standard generators of the triple cover 3.S7 are preimages C and D where CD has order 7.
Standard generators of either of the sextuple covers 6.S7 are preimages C and D where CD has order 7.


Automorphisms

An outer automorphism of A7 of order 2 may be obtained by mapping (a, b) to (a-1, b).

In the above representations, this outer automorphism is (conjugation by) c and we have d = bac.
Conversely, we have a = cd-1cd = [c, d] and b = dcd-1cdc.


Black box algorithms

To find standard generators for A7: To find standard generators for S7 = A7.2:

Presentations

Presentations for A7 and S7 = A7:2 in terms of their standard generators are given below.

< a, b | a3 = b5 = (ab)7 = (aab)2 = (ab-2ab2)2 = 1 >.

< c, d | c2 = d6 = (cd)7 = [c, d]3 = [c, dcd]2 = 1 >.


Representations

Representations are available for groups isomorphic to one of the following:

The representations of A7 available are: The representations of 2.A7 available are: The representations of 3.A7 available are:
NB: The absolutely irreducible matrix representations in characteristics 2, 5, 7 and 0 here are normalised so that (AAB)2 acts as the scalar ω. The representations of 6.A7 available are:
NB: The absolutely irreducible matrix representations in characteristics 5, 7 and 0 here are normalised so that (AAB)2 acts as the scalar -w [apart from the original one provided by J.N.Bray]. [The representations of degree 6 over GF(25) and GF(7) and degree 12 over GF(5) are ordered with respect to AABABB being in class +4A.] The representations of S7 available are: The representations of 2.S7 (plus type) available are: The representations of 2.S7 (minus type) available are: The representations of 3.S7 available are: The representations of 6.S7 (plus type) available are:

Maximal subgroups

The maximal subgroups of A7 are as follows. The maximal subgroups of S7 are as follows.

Conjugacy classes

The following are conjugacy class representatives of A7. The following are conjugacy class representatives of S7 = A7:2.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Alternating groups page Go to alternating groups page.
Old A7 page Go to old A7 page - ATLAS version 1.
ftp access Anonymous ftp access is also available on for.mat.bham.ac.uk.

Version 2.0 file created on 18th April 2000, from Version 1 file last modified on 06.01.99.
Last updated 29.11.05 by JNB.
Information checked to Level 0 on 18.04.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.