Order = 360 = 23.32.5.
Mult = 6.
Out = 22.
Porting notes
Porting incomplete. Have standard generators and representations. Most group names also copied.Standard generators
Standard generators of A6 = L2(9) = S4(2)' = M10' are a, b where a has order 2, b has order 4 and ab has order 5.
Standard generators of 2.A6 = SL2(9) are preimages A, B where AB has order 5 and ABB has order 5.
Standard generators of 3.A6 are preimages A, B where A has order 2 and B has order 4.
Standard generators of 6.A6 are preimages A, B where A has order 4, AB has order 15 and ABB has order 5.
Standard generators of S6 = A6:2a 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.S6 are preimages C, D where C has order 2 and D has order 5.
Standard generators of 3.S6 are preimages C, D where D has order 5.
Standard generators of 6.S6 are preimages C, D where C has order 2 and D has order 5.
Standard generators of PGL2(9) = A6:2b are e, f where e is in class 2D, f has order 3 and ef has order 8.
Standard generators of 2.PGL2(9) are preimages E, F where F has order 3.
Standard generators of 3.PGL2(9) are preimages E, F where EFEFF has order 5.
Standard generators of 6.PGL2(9) are preimages E, F where EFEFF has order 5. Alternatively: EFEFF has order 10 or [E,F] has order 5.
Standard generators of M10 = A6.2c 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.M10 are preimages G, H where G has order 2 and H has order 8.
Standard generators of Aut(A6) = A6.22 are i, j where i is in class 2BC, j is in class 4C and ij has order 10.
Standard generators of 3.Aut(A6) are preimages I, J where J has order 4.
Black box algorithms
Checking generators (semi-presentations)
| Group | Semi-presentation | File |
|---|---|---|
| A6 | 〈〈 a, b | some conditions 〉〉 | Download |
Presentations
| Group | Presentation | Link |
|---|---|---|
| A6 | 〈 a, b | a2 = b4 = (ab)5 = (ab2)5 = 1 〉 | Details |
| S6 | 〈 c, d | c2 = d5 = (cd)6 = [c, d]3 = [c, dcd]2 = 1 〉 | Details |
| PGL2(9) | 〈 e, f | e2 = f3 = (ef)8 = [e, f]5 = [e, fefefef−1]2 = 1 〉 | Details |
| M10 | 〈 g, h | g2 = h8 = (gh4)3 = ghghghgh−2gh3gh−2 = 1 〉 | Details |
| Aut(A6) | 〈 i, j | i2 = j4 = (ij)10 = [i, j]4 = ijij2ijij2ijij2ij−1ij2 = (ij2)4] = 1 〉 | Details |
Representations
Representations of A6
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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[b5] 8 a Std Details 0 Z[b5] 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.A6
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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(r3) Details 0 Z[z3] 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.A6
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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.A6
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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 S6
- View detailed report.
- 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.S6
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 80 Std Details 240 a Std Details 288 Std Details
Representations of 3.S6
- View detailed report.
- 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.S6
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 720 a Std Details
Representations of PGL2(9)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 10 Std Details
Representations of M10
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 10 Std Details
Representations of Aut(A6)
- View detailed report.
- Permutation representations:
Number of points ID Generators Description Link 10 Std Details
Maximal subgroups
Maximal subgroups of A6
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| A5 | 60 | 6 | Program: Generators |
| A5 | 60 | 6 | Program: Generators |
| 32:4 = F36 | 36 | 10 | Program: Generators |
| S4 | 24 | 15 | Program: Generators |
| S4 | 24 | 15 | Program: Generators |
Maximal subgroups of S6
| Subgroup | Order | Index | Programs/reps |
|---|---|---|---|
| A6 | 360 | 2 | Program: Generators |
| S5 | 120 | 6 | Program: Generators |
| S5 | 120 | 6 | Program: Generators |
| 32:D8 | 72 | 10 | Program: Generators |
| S4 × 2 | 48 | 15 | Program: Generators |
| S4 × 2 | 48 | 15 | Program: Generators |
Conjugacy classes
Conjugacy classes of A6
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 360 |
1 | |
| 2A | 8 |
a | |
| 3A | 9 |
abab−1ab2 | |
| 3B | 9 |
abab2ab−1 | |
| 4A | 4 |
b | |
| 5A | 5 |
ab | |
| 5B | 5 |
ab2 |
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Conjugacy classes of S6
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 720 |
1 | |
| 2A | 16 |
(cdcd2)2 | |
| 3A | 18 |
cdcd−1 [c,d] | |
| 3B | 18 |
cdcd | |
| 4A | 8 |
cdcd2 | |
| 5AB | 5 |
d | |
| 2B | 48 |
c | |
| 2C | 48 |
cdcdcd | |
| 4B | 8 |
cdcd−1 | |
| 6A | 6 |
cdcd2cd−1 | |
| 6B | 6 |
cd |
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Conjugacy classes of PGL2(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−1efef−1 are non−conjugate |
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Conjugacy classes of M10
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 720 |
1 | |
| 2A | 16 |
g | |
| 3AB | 9 |
gh4 | |
| 4A | 8 |
h2 | |
| 5AB | 5 |
ghgh3 | |
| 4C | 4 |
gh3 | |
| 8C | 8 |
h | |
| 8D | 8 |
h−1 |
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Conjugacy classes of Aut(A6)
| Conjugacy class | Centraliser order | Power up | Class rep(s) |
|---|---|---|---|
| 1A | 1 440 |
1 | |
| 2A | 32 |
j2 | |
| 3AB | 18 |
[i,jij] | |
| 4A | 16 |
[i,j] | |
| 5AB | 10 |
(ij)2 | |
| 2BC | 48 |
i | |
| 4B | 16 |
ij2 | |
| 6AB | 6 |
ijijij2 | |
| 2D | 40 |
(ij)5 | |
| 8AB | 8 |
ijijij−1 | |
| 10AB | 10 |
ij | |
| 4C | 8 |
j | |
| 8CD | 8 |
ijij2 |
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