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 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 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] >.
Currently, representations are available for the following decorations
of A6.
The representations of A6 available are:
- All primitive permutation representations.
-
Permutations on 6a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 6b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 10 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 15a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 15b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 2 and over GF(2).
-
Dimension 4 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 4 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 8 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 8 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 16 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- reducible over GF(4).
- All faithful irreducibles in characteristic 3 and over GF(3).
-
Dimension 3 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 3 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 4 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 6 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- reducible over GF(9).
-
Dimension 9 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 5.
-
Dimension 5 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 5 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 8 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Dimension 10 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- All faithful irreducibles in characteristic 0.
-
Dimension 5a over Z:
a and b (Magma).
-
Dimension 5b over Z:
a and b (Magma).
-
Dimension 8a over Z[b5]:
a and b (Magma).
-
Dimension 8b over Z[b5]:
a and b (Magma).
-
Dimension 9 over Z:
a and b (Magma).
-
Dimension 10 over Z:
a and b (Magma).
-
Dimension 16 over Z:
a and b (Magma).
- reducible over Q(b5).
The representations of 2.A6 available are:
-
Permutations on 80 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 144 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 240a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 240b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 2 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 2 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 4 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(9).
-
Dimension 12 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(9).
-
Dimension 4 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 4 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 10 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 10 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 20 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(25).
- Some faithful irreducibles in characteristic 0
- Dimension 4 over Z(z3):
A and B (GAP).
The representations of 3.A6 available are:
-
Permutations on 18a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 18b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 45a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 45b points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 3 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 3 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 9 over GF(4):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(4).
-
Dimension 6 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(4).
-
Dimension 18 over GF(2):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(4).
-
Dimension 3 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 15 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(25).
-
Dimension 12 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(25).
-
Dimension 30 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(25).
The representations of 6.A6 available are:
-
Permutations on 432 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 720a points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(9) [Name not fixed]:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 6 over GF(25):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 12 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(25).
-
Dimension 12 over GF(5):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- reducible over GF(25).
-
Some faithful irreducibles in characteristic 0
-
Dimension 12 over Z(z15):
A and B (GAP).
The representations of S6 = A6:2a available are:
- All faithful primitive permutation representations.
-
Permutations on 6a points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 6b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 10 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 15a points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 15b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2.S6 = 2.A6:2a available are:
-
Permutations on 80 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 240a points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 288 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 3.S6 = 3.A6:2a available are:
-
Permutations on 18a points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 18b points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 45a points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
-
Permutations on 45b points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of 6.S6 = 6.A6:2a available are:
-
Permutations on 720a points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of PGL2(9) = A6:2b available are:
-
Permutations on 10 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
The representations of M10 = A6.2c available are:
-
Permutations on 10 points:
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
The representations of Aut(A6) = A6.22 available are:
-
Permutations on 10 points:
i and
j (Meataxe),
i and
j (Meataxe binary),
i and
j (GAP).
The maximal subgroups of A6 are as follows:
-
A5, with generators
a, babb.
-
A5, with generators
a, bbab.
-
32:4 = F36, with generators
a, bababb.
-
S4, with generators
a, bbabbabab.
-
S4, with generators
a, bababbabb.
The maximal subgroups of S6 are as follows:
-
A6, with standard generators
(cdcdd)^2, cdddcd.
-
S5, with generators
c, dcdcdddd.
-
S5, with generators
cdcdcd, d.
-
32:D8, with generators
c, dcdd.
-
S4 × 2, with generators
c, dcddcddd.
-
S4 × 2, with generators
dc, cdcdcd.
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.
- 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.
Go to main ATLAS (version 2.0) page.
Go to alternating groups page.
Go to old A6 page - ATLAS version 1.
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.