# ATLAS: Alternating group A6, Linear group L2(9) Derived groups S4(2)' and M10'

Order = 360 = 23.32.5.
Mult = 6.
Out = 22.

### Standard generators

Standard generators of A6 are a and b where a has order 2, b has order 4 and ab has order 5.
In the natural representation we may take a = (1, 2)(3, 4) and b = (1, 2, 4, 5)(3, 6).
Standard generators of the double cover 2.A6 = SL2(9) are preimages A and B where AB has order 5 and ABB has order 5.
Standard generators of the triple cover 3.A6 are preimages A and B where A has order 2 and B has order 4.
Standard generators of the sextuple cover 6.A6 are preimages A and B where A has order 4, AB has order 15 and ABB has order 5.

Standard generators of S6 = A6.2a are c and d where c in class 2B/C, d has order 5 and cd has order 6 and cdd has order 6. The last condition is equivalent to cdcdddd has order 3.
In the natural representation we may take c = (1, 2) and d = (2, 3, 4, 5, 6). Alternatively, we may take c' = (1, 2)(3, 6)(4, 5) and d' = (2, 3, 4, 5, 6).
Standard generators of the double cover 2.S6 are preimages C and D where C has order 2 and D has order 5.
Standard generators of the triple cover 3.S6 are preimages C and D where D has order 5.
Standard generators of the sextuple cover 6.S6 are preimages C and D where C has order 2 and D has order 5.

Standard generators of PGL2(9) = A6.2b are e and f where e in class 2D, f has order 3 and ef has order 8.
Standard generators of either of the double covers 2.PGL2(9) are preimages E and F where F has order 3.
Standard generators of the triple cover 3.PGL2(9) are preimages E and F where EFEFF has order 5.
Standard generators of either of the sextuple covers 6.PGL2(9) are preimages E and F where F has order 3 and EFEFF has order 5 or 10. An equivalent condition to the last one is that [E, F] has order 5.

Standard generators of M10 = A6.2c are g and h where g has order 2, h has order 8, gh has order 8 and gh is conjugate to h. This last condition is equivalent to ghhhh has order 3.
Standard generators of the triple cover 3.M10 are preimages G and H where G has order 2 and H has order 8.

Standard generators of Aut(A6) = A6.22 = PGammaL2(9) are i and j where i is in class 2BC, j is in class 4C and ij has order 10.
Standard generators of the triple cover 3.Aut(A6) are preimages I and J where J has order 4.

### Presentations

Presentations of A6, S6, PGL2(9), M10 and Aut(A6) on their standard generators are given below.

< a, b | a2 = b4 = (ab)5 = (ab2)5 = 1 >.

< c, d | c2 = d5 = (cd)6 = [c, d]3 = [c, dcd]2 = 1 >.

< e, f | e2 = f3 = (ef)8 = [e, f]5 = [e, fefefef-1]2 = 1 >.

< g, h | g2 = h8 = (gh4)3 = ghghghgh-2gh3gh-2 = 1 >.

< i, j | i2 = j4 = (ij)10 = [i, j]4 = ijij2ijij2ijij2ij-1ij2 = 1 [= (ij2)4] >.

### Representations

The representations of A6 available are
• a and b as the above permutations on 6 points.
• a and b as permutations on 10 points.
• All faithful irreducibles in characteristic 2 and over GF(2).
• a and b as 4 × 4 matrices over GF(2).
• a and b as 4 × 4 matrices over GF(2).
• a and b as 8 × 8 matrices over GF(4).
• a and b as 8 × 8 matrices over GF(4).
• a and b as 16 × 16 matrices over GF(2) - reducible over GF(4).
• All faithful irreducibles in characteristic 3 and over GF(3).
• a and b as 3 × 3 matrices over GF(9).
• a and b as 3 × 3 matrices over GF(9).
• a and b as 4 × 4 matrices over GF(3).
• a and b as 6 × 6 matrices over GF(3) - reducible over GF(9).
• a and b as 9 × 9 matrices over GF(3).
• All faithful irreducibles in characteristic 5.
• a and b as 5 × 5 matrices over GF(5).
• a and b as 5 × 5 matrices over GF(5).
• a and b as 8 × 8 matrices over GF(5).
• a and b as 10 × 10 matrices over GF(5).
The representations of 3.A6 available are
• A and B as permutations on 18 points.
• A and B as permutations on 45 points.
• A and B as 3 × 3 matrices over GF(4).
The representations of S6 = A6:2a available are
• c and d as permutations on 6 points - the natural representation.
• c and d as permutations on 10 points.
• c and d as 4 × 4 matrices over GF(2) - the natural representation as Sp4(2).
The representations of 2.S6 = 2.A6:2a available are
• C and D as 4 × 4 matrices over GF(3).
• C and D as 9 × 9 matrices over GF(2).
The representations of PGL2(9) = A6:2b available are
• e and f as permutations on 10 points.
The representations of M10 = A6.2c available are
• g and h as permutations on 10 points - the natural representation as M10.
The representations of Aut(A6) = A6.22 available are
• i and j as permutations on 10 points.

### Conjugacy classes

The 7 conjugacy classes of A6 are as follows. These are with repect to the first permutation representation on 6 points with d = (2, 3, 4, 5, 6) being in class 5A (so that (1, 2, 3, 4, 5) is in class 5B) and 3-cycles being in class 3A.
• 1A: identity.
• 2A: a.
• 3A: abab-1ab2.
• 3B: abab2ab-1.
• 4A: b.
• 5A: ab.
• 5B: ab2.
The 11 conjugacy classes of S6 = A6:2a are as follows. These are with repect to the first permutation representation on 6 points with 3-cycles being in class 3A and so on.
• 1A: identity.
• 2A: (cdcd2)2.
• 3A: cdcd-1 or [c, d].
• 3B: cdcd or (cd)2.
• 4A: cdcd2.
• 5AB: d.
• 2B: c.
• 2C: cdcdcd or (cd)3.
• 4B: cdcdcd-1.
• 6A: cdcd2cd-1.
• 6B: cd.
The 11 conjugacy classes of PGL2(9) = A6:2b are as follows.
• 1A: identity.
• 2A: (ef)4.
• 3AB: f.
• 4A: (ef)2.
• 5A: .
• 5B: .
• 5A/B: [e, f] and [e, fef] are non-conjugate.
• 2D: e.
• 8A: ef.
• 8B: (ef)3.
• 10A: .
• 10B: .
• 10A/B: efefef-1 and efefef-1efef-1 in some order.
• I'll resolve the class ambiguities here when I make the 3-dimensional representation(s) for this group over GF(9).
The 8 conjugacy classes of M10 = A6.2c are as follows.
• 1A: identity.
• 2A: g.
• 3AB: gh4.
• 4A: h2.
• 5AB: ghgh3.
• 4C: gh3.
• 8C: h.
• 8D: h-1.
The 13 conjugacy classes of Aut(A6) = A6.22 are as follows.
• 1A: identity.
• 2A: j2.
• 3AB: [i, jij].
• 4A: [i, j].
• 5AB: (ij)2.
• 2BC: i.
• 4B: ij2.
• 6AB: ijijij2.
• 2D: (ij)5.
• 8AB: ijijij-1.
• 10AB: ij.
• 4C: j.
• 8CD: ijij2.