ATLAS: Symplectic group S_{6}(2)
Order = 1451520 = 2^{9}.3^{4}.5.7.
Mult = 2.
Out = 1.
Standard generators
Standard generators of S_{6}(2) are a and b where
a is in class 2A, b has order 7 and ab has order 9.
Standard generators of the double cover 2.S_{6}(2) are preimages
A and B where B has order 7 and AB has order 9.
Presentations
A presentation of S_{6}(2) on its standard generators is given below.
< a, b | a^{2} = b^{7} = (ab)^{9} = (ab^{2})^{12} = [a, b]^{3} = [a, b^{2}]^{2} = 1 >.
A shorter, and more coset enumeration friendly, presentation may be obtained by replacing (ab^{2})^{12} = 1 with [a, babab]^{2} = 1.
Representations
The representations of S_{6}(2) available are:
- All primitive permutation representations.
- a and
b as
permutations on 28 points.
- a and
b as
permutations on 36 points.
- a and
b as
permutations on 63 points.
- a and
b as
permutations on 120 points.
- a and
b as
permutations on 135 points.
- a and
b as
permutations on 315 points.
- a and
b as
permutations on 336 points.
- a and
b as
permutations on 960 points.
- All faithful irreducibles in characteristic 2.
- a and
b as
6 × 6 matrices over GF(2) - the natural representation.
- a and
b as
8 × 8 matrices over GF(2).
- a and
b as
14 × 14 matrices over GF(2).
- a and
b as
48 × 48 matrices over GF(2).
- a and
b as
64 × 64 matrices over GF(2).
- a and
b as
112 × 112 matrices over GF(2).
- a and
b as
512 × 512 matrices over GF(2) - the Steinberg representation.
- All faithful irreducibles in characteristic 3.
- a and
b as
7 × 7 matrices over GF(3).
- a and
b as
21 × 21 matrices over GF(3).
- a and
b as
27 × 27 matrices over GF(3).
- a and
b as
34 × 34 matrices over GF(3).
- a and
b as
35 × 35 matrices over GF(3).
- a and
b as
91 × 91 matrices over GF(3).
- a and
b as
98 × 98 matrices over GF(3).
- a and
b as
189 × 189 matrices over GF(3) - phi11 in the modular atlas.
- a and
b as
189 × 189 matrices over GF(3) - phi12 in the modular atlas.
- a and
b as
189 × 189 matrices over GF(3) - phi13 in the modular atlas.
- a and
b as
196 × 196 matrices over GF(3).
- a and
b as
405 × 405 matrices over GF(3).
- All faithful irreducibles in characteristic 5.
- All faithful irreducibles in characteristic 7.
- a and b as
7 × 7 matrices over Z.
- a and b as
15 × 15 matrices over Z.
- a and b as
27 × 27 matrices over Z.
- a and b as
35 × 35 matrices over Z - the deleted permutation representation.
The representations of 2.S_{6}(2) available are:
- A and
B as
permutations on 240 points. This is the representation on the cosets of a subgroup
U3(3):2 which contains outer elements of classes -4A and -8B, and the suborbits
are 1+1+112+126. Negating the outer elements of the point stabilizer would give
a different representation, with suborbits 1+1+56+56+126.
- All faithful irreducibles in characteristic 3.
- A and
B as
8 × 8 matrices over GF(3).
- A and
B as
48 × 48 matrices over GF(3).
- A and
B as
56 × 56 matrices over GF(3).
- A and
B as
56 × 56 matrices over GF(3).
- A and
B as
104 × 104 matrices over GF(3).
- A and
B as
272 × 272 matrices over GF(3).
- A and B as
8 × 8 matrices over Z.
Maximal subgroups
Return to main ATLAS page.
Last updated 31.05.00,
R.A.Wilson, R.A.Parker and J.N.Bray