# ATLAS: Alternating group A7

Order = 2520 = 23.32.5.7.
Mult = 6.
Out = 2.
Page under construction. Characteristic 0 representations have not yet been copied from Version 1.

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:
• Permutations on 7 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). the above permutations on 7 points.
• All faithful irreducibles in characteristic 2.
• All faithful irreducibles in characteristic 3 and over GF(3).
• All faithful irreducibles in characteristic 5 and over GF(5).
• All faithful irreducibles in characteristic 7.
The representations of 2.A7 available are:
• Permutations on 240 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• 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.]
• 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.]
• Dimension 4 over GF(25): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 4 over GF(25): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 8 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - reducible over GF(25).
• Dimension 14 over GF(25): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 14 over GF(25): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 20 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 20 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 28 over GF(5): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - 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.]
• Some faithful irreducibles in characteristic 0.
• Dimension 4(a) over Z(b7): A and B (GAP).
• Dimension 4(b) over Z(b7): A and B (GAP) (the complex conjugate of the preceding representation).
• Dimension 20(a) over Z(b7): A and B (GAP).
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.
• Some permutation representations.
• Permutations on 45 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 45 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 63 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Permutations on 315 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - one of many possible representations of this degree.
• All faithful irreducibles in characteristic 2 and over GF(2).
• All faithful irreducibles in characteristic 5 and over GF(5).
• All faithful irreducibles in characteristic 7.
• Some faithful irreducibles in characteristic 0
• Dimension 6 over Z[ω]: A and B (MAGMA)
• Dimension 6 over Z[ω]: A and B (GAP) (a different representation)
• Dimension 15(a) over Z[ω]: A and B (GAP)
• Dimension 15(b) over Z[ω]: A and B (GAP)
• Dimension 21(a) over Z[ω]: A and B (GAP)
• Dimension 21(b) over Z[ω]: A and B (GAP)
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.]
• Permutations on 720 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• A and B as permutations on 720 points.
• All faithful irreducibles in characteristic 5 and over GF(5).
• All faithful irreducibles in characteristic 7.
• Some faithful irreducible representations in characteristic 0.
• Dimension 6(a) over Q(r2, z3): A and B (GAP).
The representations of S7 available are:
• Permutation representations, including all faithful primitive ones.
• Permutations on 7 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 21 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 30 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - imprimitive.
• Permutations on 35 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 120 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• All faithful irreducibles in characteristic 2.
• All faithful irreducibles in characteristic 3 whose characters are printed in the ABC.
• All faithful irreducibles in characteristic 5 whose characters are printed in the ABC.
• All faithful irreducibles in characteristic 7 in the ABC in ABC order.
The representations of 2.S7 (plus type) available are:
• All faithful irreducibles in characteristic 7 and over GF(7).
• Dimension 4 over GF(49): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 8 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - reducible over GF(49).
• Dimension 16 over GF(49): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 20 over GF(49): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 28 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 32 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - reducible over GF(49).
• Dimension 40 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - reducible over GF(49).
The representations of 2.S7 (minus type) available are:
• Permutations on 240 points: C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Faithful irreducibles in characteristic 3.
• Faithful irreducibles in characteristic 5.
• Dimension 8 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 20 over GF(25): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - phi13 in the ABC.
• Dimension 20 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - phi14 in the ABC.
• Dimension 28 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 40 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - reducible over GF(25).
• Faithful irreducibles in characteristic 7.
The representations of 3.S7 available are:
• Permutations on 63 points: C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Permutations on 90 points: C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Some faithful irreducibles in characteristic 2 and over GF(2).
• All faithful irreducibles in characteristic 5.
• Dimension 6 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 12 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 30 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - phi17 in the ABC.
• Dimension 30 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - phi18 in the ABC.
• Dimension 36 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 42 over GF(5): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• All faithful irreducibles in characteristic 7.
• Dimension 12 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 18 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 30 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP).
• Dimension 42 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - phi16 in the ABC.
• Dimension 42 over GF(7): C and D (Meataxe), C and D (Meataxe binary), C and D (GAP). - phi17 in the ABC.
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.
• 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.

In order to define representations of the double cover of A7, we use the convention that AABABB is in class +4A.

In order to define the representations of the triple cover of A7, we use the convention that (ABABBBB)2 is the central element acting as scalar multiplication by z3 (as defined by the ABC).

In order to define representations of the double covers of S7, we use the convention that D is in -6C, CDCDD is in +10A, and CDD is in +12A, in both 2S7 and 2S7i. This is one of two possible choices compatible with the ABC.
(Apparently the other choice was made in version 1, but unfortunately the authors forgot to mention this fact.)

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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.
R.A.Wilson, S.J.Nickerson and J.N.Bray.