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

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

The following information is available for A6 = L2(9) = S4(2)' = M10':


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, 3, 5)(4, 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 sixfold 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 sixfold 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 sixfold covers 6.PGL2(9) are preimages E and F where F has order 3 and EFEFF has order 5 or 10 (depending on the isomorphism type of the cover). 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

Currently, representations are available for the following decorations of A6.
A6   S6   PGL2(9)   M10   A6.V4  
2.A6   2.S6  
3.A6   3.S6  
6.A6   6.S6  
The representations of A6 available are: The representations of 2.A6 available are: The representations of 3.A6 available are: The representations of 6.A6 available are: The representations of S6 = A6:2a available are: The representations of 2.S6 = 2.A6:2a available are: The representations of 3.S6 = 3.A6:2a available are: The representations of 6.S6 = 6.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

The maximal subgroups of A6 are as follows: The maximal subgroups of S6 are as follows:

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 top central element of 3.A6 and 6.A6 is (AB)5. In 2.A6 and 6.A6 B is in class -4A. 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.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Alternating groups page Go to alternating groups page.
Old A6 page Go to old A6 page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 1st November 2001.
Last updated 11.03.04 by SJN.
Information checked to Level 0 on 01.11.01 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.