ATLAS: Alternating group A_{7}
Order = 2520 = 2^{3}.3^{2}.5.7.
Mult = 6.
Out = 2.
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:
- 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.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).
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 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.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).
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.
- 7:6, with generators d, cd^2cdcd^4cd^3c.
- S_{4} × S_{3}, with generators cd^2, dcd^2cd^4cd. This subgroup contains both c and d^{2}.
[The programs for the maximal subgroups of S7 are not yet available.]
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}.
Return to main ATLAS page.
Last updated 6th January 1999,
R.A.Wilson, R.A.Parker and J.N.Bray