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}.
Standard generators
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, 4, 5)(3, 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 sextuple 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 sextuple 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 sextuple covers 6.PGL_{2}(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 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
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}] >.
Representations
The representations of A_{6} 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.A_{6} 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 S_{6} = A_{6}: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.S_{6} = 2.A_{6}: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 PGL_{2}(9) = A_{6}:2b available are
- e and
f as
permutations on 10 points.
The representations of M_{10} = A_{6}.2c available are
- g and
h as
permutations on 10 points - the natural representation as M10.
The representations of Aut(A_{6}) = A_{6}.2^{2} available are
- i and
j as
permutations on 10 points.
Maximal subgroups
Conjugacy classes
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 3-cycles being in class 3A.
- 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 3-cycles 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 non-conjugate.
- 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 3-dimensional 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}.
Return to main ATLAS page.
Last updated 14th October 1999,
R.A.Wilson, R.A.Parker and J.N.Bray