# ATLAS: Symplectic group S6(2)

Order = 1451520 = 29.34.5.7.
Mult = 2.
Out = 1.

### Standard generators

Standard generators of S6(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.S6(2) are preimages A and B where B has order 7 and AB has order 9.

### Presentations

A presentation of S6(2) on its standard generators is given below.

< a, b | a2 = b7 = (ab)9 = (ab2)12 = [a, b]3 = [a, b2]2 = 1 >.

A shorter, and more coset enumeration friendly, presentation may be obtained by replacing (ab2)12 = 1 with [a, babab]2 = 1.

### Representations

The representations of S6(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.S6(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.