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

Representations of 2.A6

Representations of 3.A6

Representations of 6.A6

Representations of S6

Representations of 2.S6

Representations of 3.S6

Representations of 6.S6

Representations of PGL2(9)

Representations of M10

Representations of Aut(A6)


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)
1A360 1
2A8 a
3A9 abab−1ab2
3B9 abab2ab−1
4A4 b
5A5 ab
5B5 ab2

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Conjugacy classes of S6

Conjugacy class Centraliser order Power up Class rep(s)
1A720 1
2A16 (cdcd2)2
3A18 cdcd−1
[c,d]
3B18 cdcd
4A8 cdcd2
5AB5 d
2B48 c
2C48 cdcdcd
4B8 cdcd−1
6A6 cdcd2cd−1
6B6 cd

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Conjugacy classes of PGL2(9)

Conjugacy class Centraliser order Power up Class rep(s)
1A720 1
2A16 (ef)4
3AB9 f
4A8 (ef)2
5A10 ? See 5A/B below
5B10 ? See 5A/B below
2D20 e
8A8 ef
8B8 (ef)3
10A10 ? See 10A/B
10B10 ? See 10A/B
5A/B [e,f] and [e,fef] are nonconjugate
10A/B efefef−1 and efefef−1efef−1 are nonconjugate

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Conjugacy classes of M10

Conjugacy class Centraliser order Power up Class rep(s)
1A720 1
2A16 g
3AB9 gh4
4A8 h2
5AB5 ghgh3
4C4 gh3
8C8 h
8D8 h−1

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Conjugacy classes of Aut(A6)

Conjugacy class Centraliser order Power up Class rep(s)
1A1 440 1
2A32 j2
3AB18 [i,jij]
4A16 [i,j]
5AB10 (ij)2
2BC48 i
4B16 ij2
6AB6 ijijij2
2D40 (ij)5
8AB8 ijijij−1
10AB10 ij
4C8 j
8CD8 ijij2

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