# 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:
• Find an element of order 6. This squares to x in class 3A.
[The probability of success at each attempt is 1 in 12.]
• Find an element y of order 5.
[The probability of success at each attempt is 1 in 5.]
• Find conjugates a of x and b of y such that ab has order 7.
[The probability of success at each attempt is 1 in 7.]
To find standard generators for S7 = A7.2:
• Find an element of order 10. This powers up to x in class 2B.
[The probability of success at each attempt is 1 in 10 (or 1 in 5 if you look through outer elements only).]
• Find an element y of order 7.
[The probability of success at each attempt is 1 in 7 (or 2 in 7 if you look through inner elements only).]
• Find conjugates c of x and z of y such that cz has order 6.
[The probability of success at each attempt is 1 in 3.]
• Now c and d = zc are standard generators of S7.

### 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:

 A7. 2.A7. 3.A7. 6.A7. S7. 2.S7 (+)    and    2.S7 (-). 3.S7. 6.S7 (+).
The representations of A7 available are:

• a and b as the above permutations on 7 points.
• All faithful irreducibles in characteristic 2.
• a and b as 4 × 4 matrices over GF(2).
• a and b as 4 × 4 matrices over GF(2) - the dual of the above.
• a and b as 6 × 6 matrices over GF(2).
• a and b as 14 × 14 matrices over GF(2).
• a and b as 20 × 20 matrices over GF(2).
• All faithful irreducibles in characteristic 3 and over GF(3).
• a and b as 6 × 6 matrices over GF(3).
• a and b as 10 × 10 matrices over GF(9).
• a and b as 10 × 10 matrices over GF(9).
• a and b as 13 × 13 matrices over GF(3).
• a and b as 15 × 15 matrices over GF(3).
• a and b as 20 × 20 matrices over GF(3) - reducible over GF(9).
• All faithful irreducibles in characteristic 5 and over GF(5).
• a and b as 6 × 6 matrices over GF(5).
• a and b as 8 × 8 matrices over GF(5).
• a and b as 10 × 10 matrices over GF(25).
• a and b as 10 × 10 matrices over GF(25).
• a and b as 13 × 13 matrices over GF(5).
• a and b as 15 × 15 matrices over GF(5).
• a and b as 20 × 20 matrices over GF(5) - reducible over GF(25).
• a and b as 35 × 35 matrices over GF(5).
• All faithful irreducibles in characteristic 7.
• a and b as 5 × 5 matrices over GF(7).
• a and b as 10 × 10 matrices over GF(7).
• a and b as 14 × 14 matrices over GF(7).
• a and b as 14 × 14 matrices over GF(7).
• a and b as 21 × 21 matrices over GF(7).
• a and b as 35 × 35 matrices over GF(7).
The representations of 2.A7 available are:
• A and B as permutations on 240 points.
• All faithful irreducibles in characteristic 3 and over GF(3).
[The representations of degree 6 are ordered with respect to AABABB being in class +4A.]
• A and B as 4 × 4 matrices over GF(9).
• A and B as 4 × 4 matrices over GF(9).
• A and B as 6 × 6 matrices over GF(9).
• A and B as 6 × 6 matrices over GF(9).
• A and B as 8 × 8 matrices over GF(3) - reducible over GF(9).
• A and B as 12 × 12 matrices over GF(3) - reducible over GF(9).
• A and B as 36 × 36 matrices over GF(3).
• All faithful irreducibles in characteristic 5 and over GF(5).
[The representations of degree 14 are ordered with respect to AABABB being in class +4A.]
• A and B as 4 × 4 matrices over GF(25).
• A and B as 4 × 4 matrices over GF(25).
• A and B as 8 × 8 matrices over GF(5) - reducible over GF(25).
• A and B as 14 × 14 matrices over GF(25).
• A and B as 14 × 14 matrices over GF(25).
• A and B as 20 × 20 matrices over GF(5).
• A and B as 20 × 20 matrices over GF(5).
• A and B as 28 × 28 matrices over GF(5) - reducible over GF(25).
• All faithful irreducibles in characteristic 7.
[The representations of degree 14 are ordered with respect to AABABB being in class +4A.]
• A and B as 4 × 4 matrices over GF(7).
• A and B as 14 × 14 matrices over GF(7).
• A and B as 14 × 14 matrices over GF(7).
• A and B as 16 × 16 matrices over GF(7).
• A and B as 20 × 20 matrices over GF(7).
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).
• A and B as permutations on 45 points.
• A and B as permutations on 45 points.
• A and B as permutations on 63 points.
• A and B as permutations on 315 points - one of many possible representations of this degree.
• A and B as 6 × 6 matrices over GF(4).
• A and B as 15 × 15 matrices over GF(4).
• A and B as 24 × 24 matrices over GF(4).
• A and B as 24 × 24 matrices over GF(4).
• A and B as 12 × 12 matrices over GF(2).
• A and B as 30 × 30 matrices over GF(2).
• A and B as 48 × 48 matrices over GF(2).
• A and B as 48 × 48 matrices over GF(2).
• A and B as 3 × 3 matrices over GF(25).
• A and B as 6 × 6 matrices over GF(25).
• A and B as 15 × 15 matrices over GF(25).
• A and B as 15 × 15 matrices over GF(25).
• A and B as 18 × 18 matrices over GF(25).
• A and B as 21 × 21 matrices over GF(25).
• A and B as 6 × 6 matrices over GF(5).
• A and B as 12 × 12 matrices over GF(5).
• A and B as 30 × 30 matrices over GF(5).
• A and B as 30 × 30 matrices over GF(5).
• A and B as 36 × 36 matrices over GF(5).
• A and B as 42 × 42 matrices over GF(5).
• A and B as 6 × 6 matrices over GF(7).
• A and B as 9 × 9 matrices over GF(7).
• A and B as 15 × 15 matrices over GF(7).
• A and B as 21 × 21 matrices over GF(7).
• A and B as 21 × 21 matrices over GF(7).
• A and B as 6 × 6 matrices over Z[w].
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.]
• A and B as permutations on 720 points.
• A and B as 6 × 6 matrices over GF(25) - phi21 in ABC.
• A and B as 6 × 6 matrices over GF(25) - phi22 in ABC.
• A and B as 12 × 12 matrices over GF(25).
• A and B as 24 × 24 matrices over GF(25).
• A and B as 12 × 12 matrices over GF(5) - phi21 + follower in ABC.
• A and B as 12 × 12 matrices over GF(5) - phi22 + follower in ABC.
• A and B as 24 × 24 matrices over GF(5).
• A and B as 48 × 48 matrices over GF(5).
• A and B as 6 × 6 matrices over GF(7).
• A and B as 6 × 6 matrices over GF(7).
• A and B as 6 × 6 matrices over GF(7) - kindly provided by J.N.Bray [Dual of the one immediately above, and automorph of the one just above that].
• A and B as 24 × 24 matrices over GF(7).
The representations of S7 available are:
• Permutation representations, including all faithful primitive ones.
• c and d as the above permutations on 7 points.
• c and d as permutations on 21 points.
• c and d as permutations on 30 points - imprimitive.
• c and d as permutations on 35 points.
• c and d as permutations on 120 points.
• All faithful irreducibles in characteristic 2.
• c and d as 6 × 6 matrices over GF(2).
• c and d as 8 × 8 matrices over GF(2).
• c and d as 14 × 14 matrices over GF(2).
• c and d as 20 × 20 matrices over GF(2).
• All faithful irreducibles in characteristic 3 whose characters are printed in the ABC.
• c and d as 6 × 6 matrices over GF(3).
• c and d as 13 × 13 matrices over GF(3).
• c and d as 15 × 15 matrices over GF(3).
• c and d as 20 × 20 matrices over GF(3).
• All faithful irreducibles in characteristic 5 whose characters are printed in the ABC.
• c and d as 6 × 6 matrices over GF(5).
• c and d as 8 × 8 matrices over GF(5).
• c and d as 13 × 13 matrices over GF(5).
• c and d as 15 × 15 matrices over GF(5).
• c and d as 20 × 20 matrices over GF(5).
• c and d as 35 × 35 matrices over GF(5).
• All faithful irreducibles in characteristic 7 in the ABC in ABC order.
• c and d as 5 × 5 matrices over GF(7).
• c and d as 10 × 10 matrices over GF(7).
• c and d as 14 × 14 matrices over GF(7).
• c and d as 14 × 14 matrices over GF(7).
• c and d as 21 × 21 matrices over GF(7).
• c and d as 35 × 35 matrices over GF(7).
The representations of 2.S7 (plus type) available are:
• All faithful irreducibles in characteristic 7 and over GF(7).
• C and D as 4 × 4 matrices over GF(49).
• C and D as 8 × 8 matrices over GF(7) - reducible over GF(49).
• C and D as 16 × 16 matrices over GF(49).
• C and D as 20 × 20 matrices over GF(49).
• C and D as 28 × 28 matrices over GF(7).
• C and D as 32 × 32 matrices over GF(7) - reducible over GF(49).
• C and D as 40 × 40 matrices over GF(7) - reducible over GF(49).
The representations of 2.S7 (minus type) available are:
• C and D as permutations on 240 points.
• C and D as 8 × 8 matrices over GF(3).
• C and D as 12 × 12 matrices over GF(3).
• C and D as 36 × 36 matrices over GF(3).
• C and D as 8 × 8 matrices over GF(5).
• C and D as 28 × 28 matrices over GF(5).
• C and D as 20 × 20 matrices over GF(5).
• C and D as 20 × 20 matrices over GF(25).
• C and D as 40 × 40 matrices over GF(5) - reducible over GF(25).
• C and D as 4 × 4 matrices over GF(7).
• C and D as 16 × 16 matrices over GF(7).
• C and D as 20 × 20 matrices over GF(7).
• C and D as 28 × 28 matrices over GF(7).
The representations of 3.S7 available are:
• C and D as permutations on 63 points.
• C and D as permutations on 90 points.
• All faithful irreducibles in characteristic 2 and over GF(2).
• C and D as 12 × 12 matrices over GF(2).
• C and D as 30 × 30 matrices over GF(2).
• C and D as 48 × 48 matrices over GF(4) - not yet available.
• C and D as 48 × 48 matrices over GF(4) - not yet available.
• C and D as 96 × 96 matrices over GF(2) - reducible over GF(4).
• All faithful irreducibles in characteristic 5.
• C and D as 6 × 6 matrices over GF(5).
• C and D as 12 × 12 matrices over GF(5).
• C and D as 30 × 30 matrices over GF(5) - phi17 in the ABC.
• C and D as 30 × 30 matrices over GF(5) - phi18 in the ABC.
• C and D as 36 × 36 matrices over GF(5).
• C and D as 42 × 42 matrices over GF(5).
• All faithful irreducibles in characteristic 7.
• C and D as 12 × 12 matrices over GF(7).
• C and D as 18 × 18 matrices over GF(7).
• C and D as 30 × 30 matrices over GF(7).
• C and D as 42 × 42 matrices over GF(7) - phi16 in the ABC.
• C and D as 42 × 42 matrices over GF(7) - phi17 in the ABC.
The representations of 6.S7 (plus type) available are:
• C and D as 12 × 12 matrices over GF(7).
• C and D as 48 × 48 matrices over GF(7).

### 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.
• 1A: identity.
• 2A: ab-1ab.
• 3A: a.
• 3B: a-1bab.
• 4A: a-1bab2.
• 5A: b.
• 6A: ababab2.
• 7A: ab.
• 7B: a-1b.
The following are conjugacy class representatives of S7 = A7:2.
• 1A: identity.
• 2A: (cd3)3.
• 3A: cdcd-1 or [c, d] or (cd3)2.
• 3B: d2.
• 4A: cdcd2cd-2.
• 5A: cdcdcd-1.
• 6A: cd3.
• 7AB: cd.
• 2B: c.
• 2C: d3.
• 4B: cdcdcd-2 or (cd2)3.
• 6B: cd2cd2cd-2.
• 6C: d.
• 10A: cdcd2.
• 12A: cd2.

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Last updated 6th January 1999,
R.A.Wilson, R.A.Parker and J.N.Bray