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 The representations of 3.A6 available are The representations of S6 = A6:2a available are The representations of 2.S6 = 2.A6:2a available are The representations of PGL2(9) = A6:2b available are The representations of M10 = A6.2c available are The representations of Aut(A6) = A6.22 available are

Maximal subgroups


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. 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. The 11 conjugacy classes of PGL2(9) = A6:2b are as follows. The 8 conjugacy classes of M10 = A6.2c are as follows. The 13 conjugacy classes of Aut(A6) = A6.22 are as follows.
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Last updated 14th October 1999,
R.A.Wilson, R.A.Parker and J.N.Bray