ATLAS: Alternating group A_{6},
Linear group L_{2}(9)
Derived groups S_{4}(2)' and M_{10}'
Order = 360 = 2^{3}.3^{2}.5.
Mult = 6.
Out = 2^{2}.
The following information is available for A_{6} = L_{2}(9) =
S_{4}(2)' = M_{10}':
Standard generators of A_{6} 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.A_{6} = SL_{2}(9)
are preimages A and B where AB has order 5 and
ABB has order 5.
Standard generators of the triple cover 3.A_{6} are preimages A and B where A has order 2 and B has order 4.
Standard generators of the sixfold cover 6.A_{6} are preimages
A and B where A has order 4, AB has order 15
and ABB has order 5.
Standard generators of S_{6} = A_{6}.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.S_{6} are preimages
C and D where C has order 2 and D has order 5.
Standard generators of the triple cover 3.S_{6} are preimages
C and D where D has order 5.
Standard generators of the sixfold cover 6.S_{6} are preimages
C and D where C has order 2 and D has order 5.
Standard generators of PGL_{2}(9) = A_{6}.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.PGL_{2}(9) are
preimages E and F where F has order 3.
Standard generators of the triple cover 3.PGL_{2}(9) are preimages
E and F where EFEFF has order 5.
Standard generators of either of the sixfold covers 6.PGL_{2}(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 M_{10} = A_{6}.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.M_{10} are preimages
G and H where G has order 2 and H has order 8.
Standard generators of Aut(A_{6}) = A_{6}.2^{2} =
PGammaL_{2}(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(A_{6}) are preimages I and J where J has order 4.
Presentations of A_{6}, S_{6}, PGL_{2}(9), M_{10} and Aut(A_{6}) on their standard generators are given below.
< a, b  a^{2} = b^{4} = (ab)^{5} = (ab^{2})^{5} = 1 >.
< c, d  c^{2} = d^{5} = (cd)^{6} = [c, d]^{3} = [c, dcd]^{2} = 1 >.
< e, f  e^{2} = f^{3} = (ef)^{8} = [e, f]^{5} = [e, fefefef^{1}]^{2} = 1 >.
< g, h  g^{2} = h^{8} = (gh^{4})^{3} = ghghghgh^{2}gh^{3}gh^{2} = 1 >.
< i, j  i^{2} = j^{4} = (ij)^{10} = [i, j]^{4} = ijij^{2}ijij^{2}ijij^{2}ij^{1}ij^{2} = 1 [= (ij^{2})^{4}] >.
Currently, representations are available for the following decorations
of A_{6}.
The representations of A_{6} 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.A_{6} 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.A_{6} 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.A_{6} 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 S_{6} = A_{6}: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.S_{6} = 2.A_{6}: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.S_{6} = 3.A_{6}: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.S_{6} = 6.A_{6}:2a available are:

Permutations on 720a points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The representations of PGL_{2}(9) = A_{6}:2b available are:

Permutations on 10 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
The representations of M_{10} = A_{6}.2c available are:

Permutations on 10 points:
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
The representations of Aut(A_{6}) = A_{6}.2^{2} available are:

Permutations on 10 points:
i and
j (Meataxe),
i and
j (Meataxe binary),
i and
j (GAP).
The maximal subgroups of A_{6} are as follows:

A_{5}, with generators
a, babb.

A_{5}, with generators
a, bbab.

3^{2}:4 = F_{36}, with generators
a, bababb.

S_{4}, with generators
a, bbabbabab.

S_{4}, with generators
a, bababbabb.
The maximal subgroups of S_{6} are as follows:

A_{6}, with standard generators
(cdcdd)^2, cdddcd.

S_{5}, with generators
c, dcdcdddd.

S_{5}, with generators
cdcdcd, d.

3^{2}:D_{8}, with generators
c, dcdd.

S_{4} × 2, with generators
c, dcddcddd.

S_{4} × 2, with generators
dc, cdcdcd.
The 7 conjugacy classes of A_{6} 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 3cycles 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^{1}ab^{2}.
 3B: abab^{2}ab^{1}.
 4A: b.
 5A: ab.
 5B: ab^{2}.
The 11 conjugacy classes of S_{6} = A_{6}:2a are as follows. These are with repect to the first permutation representation on 6 points with 3cycles being in class 3A and so on.
 1A: identity.
 2A: (cdcd^{2})^{2}.
 3A: cdcd^{1} or [c, d].
 3B: cdcd or (cd)^{2}.
 4A: cdcd^{2}.
 5AB: d.
 2B: c.
 2C: cdcdcd or (cd)^{3}.
 4B: cdcdcd^{1}.
 6A: cdcd^{2}cd^{1}.
 6B: cd.
The 11 conjugacy classes of PGL_{2}(9) = A_{6}:2b are as follows.
 1A: identity.
 2A: (ef)^{4}.
 3AB: f.
 4A: (ef)^{2}.
 5A: .
 5B: .
 5A/B: [e, f] and [e, fef] are nonconjugate.
 2D: e.
 8A: ef.
 8B: (ef)^{3}.
 10A: .
 10B: .
 10A/B: efefef^{1} and efefef^{1}efef^{1} in some order.
 I'll resolve the class ambiguities here when I make the 3dimensional representation(s) for this group over GF(9).
The 8 conjugacy classes of M_{10} = A_{6}.2c are as follows.
 1A: identity.
 2A: g.
 3AB: gh^{4}.
 4A: h^{2}.
 5AB: ghgh^{3}.
 4C: gh^{3}.
 8C: h.
 8D: h^{1}.
The 13 conjugacy classes of Aut(A_{6}) = A_{6}.2^{2} are as follows.
 1A: identity.
 2A: j^{2}.
 3AB: [i, jij].
 4A: [i, j].
 5AB: (ij)^{2}.
 2BC: i.
 4B: ij^{2}.
 6AB: ijijij^{2}.
 2D: (ij)^{5}.
 8AB: ijijij^{1}.
 10AB: ij.
 4C: j.
 8CD: ijij^{2}.
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.