Order = 360 = 2^{3}.3^{2}.5.
Mult = 6.
Out = 2^{2}.
Porting notes
Porting incomplete. Have standard generators and representations. Most group names also copied.Standard generators
Standard generators of A_{6} = L_{2}(9) = S_{4}(2)' = M_{10}' are a, b where a has order 2, b has order 4 and ab has order 5.
Standard generators of 2.A_{6} = SL_{2}(9) are preimages A, B where AB has order 5 and ABB has order 5.
Standard generators of 3.A_{6} are preimages A, B where A has order 2 and B has order 4.
Standard generators of 6.A_{6} are preimages A, B where A has order 4, AB has order 15 and ABB has order 5.
Standard generators of S_{6} = A_{6}:2a are c, d where c is in class 2B/C, d has order 5, cd has order 6 and cdd has order 6. Alternatively: c is in class 2B/C, d has order 5, cd has order 6 and cddddd has order 3.
Standard generators of 2.S_{6} are preimages C, D where C has order 2 and D has order 5.
Standard generators of 3.S_{6} are preimages C, D where D has order 5.
Standard generators of 6.S_{6} are preimages C, D where C has order 2 and D has order 5.
Standard generators of PGL_{2}(9) = A_{6}:2b are e, f where e is in class 2D, f has order 3 and ef has order 8.
Standard generators of 2.PGL_{2}(9) are preimages E, F where F has order 3.
Standard generators of 3.PGL_{2}(9) are preimages E, F where EFEFF has order 5.
Standard generators of 6.PGL_{2}(9) are preimages E, F where EFEFF has order 5. Alternatively: EFEFF has order 10 or [E,F] has order 5.
Standard generators of M_{10} = A_{6}.2c are g, h where g has order 2, h has order 8, gh has order 8 and ghhhh has order 3.
Standard generators of 3.M_{10} are preimages G, H where G has order 2 and H has order 8.
Standard generators of Aut(A_{6}) = A_{6}.2^{2} are i, j where i is in class 2BC, j is in class 4C and ij has order 10.
Standard generators of 3.Aut(A_{6}) are preimages I, J where J has order 4.
Black box algorithms
Checking generators (semi-presentations)
Group | Semi-presentation | File |
---|---|---|
A_{6} | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
Group | Presentation | Link |
---|---|---|
A_{6} | 〈 a, b | a^{2} = b^{4} = (ab)^{5} = (ab^{2})^{5} = 1 〉 | Details |
S_{6} | 〈 c, d | c^{2} = d^{5} = (cd)^{6} = [c, d]^{3} = [c, dcd]^{2} = 1 〉 | Details |
PGL_{2}(9) | 〈 e, f | e^{2} = f^{3} = (ef)^{8} = [e, f]^{5} = [e, fefefef^{−1}]^{2} = 1 〉 | Details |
M_{10} | 〈 g, h | g^{2} = h^{8} = (gh^{4})^{3} = ghghghgh^{−2}gh^{3}gh^{−2} = 1 〉 | Details |
Aut(A_{6}) | 〈 i, j | i^{2} = j^{4} = (ij)^{10} = [i, j]^{4} = ijij^{2}ijij^{2}ijij^{2}ij^{−1}ij^{2} = (ij^{2})^{4}] = 1 〉 | Details |
Representations
Representations of A_{6}
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- Permutation representations:
Number of points ID Generators Description Link 6 a Std Details 6 b Std Details 10 Std Details 15 a Std Details 15 b Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 Z 5 a Std Details 0 Z 5 b Std Details 0 C 6 a Std Details 0 Z[b_{5}] 8 a Std Details 0 Z[b_{5}] 8 b Std Details 0 Z 9 Std Details 0 Z 10 Std Details 0 C 12 a Std Details 0 Z 16 Std Details Char Ring Dimension ID Generators Description Link 2 GF(2) 4 a Std Details 2 GF(2) 4 b Std Details 2 GF(4) 8 a Std Details 2 GF(4) 8 b Std Details 2 GF(2) 16 Std Details Char Ring Dimension ID Generators Description Link 3 GF(9) 3 a Std Details 3 GF(9) 3 b Std Details 3 GF(3) 4 Std Details 3 GF(3) 6 Std Details 3 GF(3) 9 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 5 a Std Details 5 GF(5) 5 b Std Details 5 GF(5) 8 Std Details 5 GF(5) 10 Std Details
Representations of 2.A_{6}
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- Permutation representations:
Number of points ID Generators Description Link 80 Std Details 144 Std Details 240 a Std Details 240 b Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 H 2 a Std Over quaternions over Q(r_{3}) Details 0 Z[z_{3}] 4 Std Details 0 Z[ω] 4 a Std Details Char Ring Dimension ID Generators Description Link 3 GF(9) 2 a Std Details 3 GF(9) 2 b Std Details 3 GF(3) 4 Std Details 3 GF(9) 6 a Std Details 3 GF(9) 6 b Std Details 3 GF(3) 12 Std Details Char Ring Dimension ID Generators Description Link 5 GF(5) 4 a Std Details 5 GF(5) 4 b Std Details 5 GF(25) 10 a Std Details 5 GF(25) 10 b Std Details 5 GF(5) 20 Std Details
Representations of 3.A_{6}
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- Permutation representations:
Number of points ID Generators Description Link 18 a Std Details 18 b Std Details 45 a Std Details 45 b Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 C 3 a Std Details 0 Z[ω] 6 Std Details 0 Z[ω] 9 Std Details 0 C 15 Std Details Char Ring Dimension ID Generators Description Link 2 GF(4) 3 a Std Details 2 GF(4) 3 b Std Details 2 GF(2) 6 a Std Details 2 GF(2) 6 b Std Details 2 GF(4) 9 a Std Details 2 GF(2) 18 Std Details Char Ring Dimension ID Generators Description Link 5 GF(25) 3 a Std Details 5 GF(25) 6 a Std Details 5 GF(5) 6 b Std Details 5 GF(5) 12 Std Details 5 GF(25) 15 a Std Details 5 GF(5) 30 Std Details
Representations of 6.A_{6}
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- Permutation representations:
Number of points ID Generators Description Link 432 Std Details 720 a Std Details - Matrix representations
Char Ring Dimension ID Generators Description Link 0 A 12 Std Details 3 GF(9) 6 a Std Details 5 GF(25) 6 a Std Details 5 GF(25) 6 b Std Details 5 GF(5) 12 a Std Details 5 GF(5) 12 b Std Details
Representations of S_{6}
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- Permutation representations:
Number of points ID Generators Description Link 6 a Std Details 6 b Std Details 10 Std Details 15 a Std Details 15 b Std Details
Representations of 2.S_{6}
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- Permutation representations:
Number of points ID Generators Description Link 80 Std Details 240 a Std Details 288 Std Details
Representations of 3.S_{6}
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- Permutation representations:
Number of points ID Generators Description Link 18 a Std Details 18 b Std Details 45 a Std Details 45 b Std Details
Representations of 6.S_{6}
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 720 a Std Details
Representations of PGL_{2}(9)
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- Permutation representations:
Number of points ID Generators Description Link 10 Std Details
Representations of M_{10}
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- Permutation representations:
Number of points ID Generators Description Link 10 Std Details
Representations of Aut(A_{6})
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- Permutation representations:
Number of points ID Generators Description Link 10 Std Details
Maximal subgroups
Maximal subgroups of A_{6}
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
A5 | 60 | 6 | Program: Generators |
A5 | 60 | 6 | Program: Generators |
3^{2}:4 = F_{36} | 36 | 10 | Program: Generators |
S4 | 24 | 15 | Program: Generators |
S4 | 24 | 15 | Program: Generators |
Maximal subgroups of S_{6}
Subgroup | Order | Index | Programs/reps |
---|---|---|---|
A6 | 360 | 2 | Program: Generators |
S5 | 120 | 6 | Program: Generators |
S5 | 120 | 6 | Program: Generators |
3^{2}:D_{8} | 72 | 10 | Program: Generators |
S_{4} × 2 | 48 | 15 | Program: Generators |
S_{4} × 2 | 48 | 15 | Program: Generators |
Conjugacy classes
Conjugacy classes of A_{6}
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 360 |
1 | |
2A | 8 |
a | |
3A | 9 |
abab^{−1}ab^{2} | |
3B | 9 |
abab^{2}ab^{−1} | |
4A | 4 |
b | |
5A | 5 |
ab | |
5B | 5 |
ab^{2} |
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Conjugacy classes of S_{6}
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 720 |
1 | |
2A | 16 |
(cdcd^{2})^{2} | |
3A | 18 |
cdcd^{−1} [c,d] | |
3B | 18 |
cdcd | |
4A | 8 |
cdcd^{2} | |
5AB | 5 |
d | |
2B | 48 |
c | |
2C | 48 |
cdcdcd | |
4B | 8 |
cdcd^{−1} | |
6A | 6 |
cdcd^{2}cd^{−1} | |
6B | 6 |
cd |
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Conjugacy classes of PGL_{2}(9)
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 720 |
1 | |
2A | 16 |
(ef)^{4} | |
3AB | 9 |
f | |
4A | 8 |
(ef)^{2} | |
5A | 10 |
? See 5A/B below | |
5B | 10 |
? See 5A/B below | |
2D | 20 |
e | |
8A | 8 |
ef | |
8B | 8 |
(ef)^{3} | |
10A | 10 |
? See 10A/B | |
10B | 10 |
? See 10A/B | |
5A/B |
[e,f] and [e,fef] are non−conjugate | ||
10A/B |
efefef^{−1} and efefef^{−1}efef^{−1} are non−conjugate |
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Conjugacy classes of M_{10}
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 720 |
1 | |
2A | 16 |
g | |
3AB | 9 |
gh^{4} | |
4A | 8 |
h^{2} | |
5AB | 5 |
ghgh^{3} | |
4C | 4 |
gh^{3} | |
8C | 8 |
h | |
8D | 8 |
h^{−1} |
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Conjugacy classes of Aut(A_{6})
Conjugacy class | Centraliser order | Power up | Class rep(s) |
---|---|---|---|
1A | 1 440 |
1 | |
2A | 32 |
j^{2} | |
3AB | 18 |
[i,jij] | |
4A | 16 |
[i,j] | |
5AB | 10 |
(ij)^{2} | |
2BC | 48 |
i | |
4B | 16 |
ij^{2} | |
6AB | 6 |
ijijij^{2} | |
2D | 40 |
(ij)^{5} | |
8AB | 8 |
ijijij^{−1} | |
10AB | 10 |
ij | |
4C | 8 |
j | |
8CD | 8 |
ijij^{2} |
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