ATLAS: Alternating group A_{7}
Order = 2520 = 2^{3}.3^{2}.5.7.
Mult = 6.
Out = 2.
See also ATLAS of Finite Groups p10
The following information is available for A_{7}:
Standard generators
Standard generators of A_{7} 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.A_{7} 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.A_{7} are preimages A and B where B has order 5 and AB has order 7.
Standard generators of the sextuple cover 6.A_{7} are preimages
A and B where B has order 5 and AB has order 7.
Standard generators of S_{7} 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.S_{7} are
preimages C and D where CD has order 7.
Standard generators of the triple cover 3.S_{7} are preimages C
and D where CD has order 7.
Standard generators of either of the sextuple covers 6.S_{7} are
preimages C and D where CD has order 7.
Automorphisms
An outer automorphism of A_{7} 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^{1}cd = [c, d] and
b = dcd^{1}cdc.
Black box algorithms
To find standard generators for A_{7}:
 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 S_{7} = A_{7}.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 S_{7}.
Presentations
Presentations for A_{7} and S_{7} = A_{7}:2 in terms of their standard generators are given below.
< a, b  a^{3} = b^{5} = (ab)^{7} = (aa^{b})^{2} = (ab^{2}ab^{2})^{2} = 1 >.
< c, d  c^{2} = d^{6} = (cd)^{7} = [c, d]^{3} = [c, dcd]^{2} = 1 >.
Representations
Representations are available for groups isomorphic to one of the following:
The representations of A_{7} 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.

Dimension 4 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 4 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 6 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 3 and over GF(3).

Dimension 6 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 13 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(9).
 All faithful irreducibles in characteristic 5 and over GF(5).

Dimension 6 over GF(5):
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).

Dimension 10 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 13 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15 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).
 reducible over GF(25).

Dimension 35 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 7.

Dimension 5 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 21 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.A_{7} 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).
 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.A_{7} available are:
NB: The absolutely irreducible matrix representations in characteristics 2, 5, 7 and 0 here are normalised so that (AA^{B})^{2} acts as the scalar ω.
 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).
 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.A_{7} available are:
NB: The absolutely irreducible matrix representations in characteristics 5, 7 and 0 here are normalised so that (AA^{B})^{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).
 Some faithful irreducible representations in characteristic 0.
 Dimension 6(a) over Q(r2, z3):
A and B (GAP).
The representations of S_{7} 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.S_{7} (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.S_{7} (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.S_{7} 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.S_{7} (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 A_{7} are as follows.
The maximal subgroups of S_{7} are as follows.
 A_{7}, with standard generators cd^5cd, dcd^5cdc.
 S_{6}, with standard generators c, cdcdcd^5.
 S_{5} × 2, with generators c, cdcd^2cd^4 mapping onto standard generators of S5.
 S_{4} × S_{3}, with generators cd^2, dcd^2cd^4cd. This subgroup contains both c and d^{2}.
 7:6, with generators d, cd^2cdcd^4cd^3c.
Conjugacy classes
The following are conjugacy class representatives of A_{7}.
 1A: identity.
 2A: ab^{1}ab.
 3A: a.
 3B: a^{1}bab.
 4A: a^{1}bab^{2}.
 5A: b.
 6A: ababab^{2}.
 7A: ab.
 7B: a^{1}b.
The following are conjugacy class representatives of S_{7} = A_{7}:2.
 1A: identity.
 2A: (cd^{3})^{3}.
 3A: cdcd^{1} or [c, d] or (cd^{3})^{2}.
 3B: d^{2}.
 4A: cdcd^{2}cd^{2}.
 5A: cdcdcd^{1}.
 6A: cd^{3}.
 7AB: cd.
 2B: c.
 2C: d^{3}.
 4B: cdcdcd^{2} or (cd^{2})^{3}.
 6B: cd^{2}cd^{2}cd^{2}.
 6C: d.
 10A: cdcd^{2}.
 12A: cd^{2}.
Go to main ATLAS (version 2.0) page.
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Go to old A7 page  ATLAS version 1.
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.