ATLAS: Alternating group A7

Order = 2520 = 23.32.5.7.
Mult = 6.
Out = 2.

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 w (omega). 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. [The programs for the maximal subgroups of S7 are not yet available.]

Conjugacy classes

The following are conjugacy class representatives of A7. The following are conjugacy class representatives of S7 = A7:2.
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Last updated 6th January 1999,
R.A.Wilson, R.A.Parker and J.N.Bray