# ATLAS: Symplectic group S6(2)

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

The following information is available for S6(2):

### 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.

This presentation, and that of the covering group 2.S6(2), is available in Magma format as follows: S6(2) on a and b and 2S6(2) on A and B.

### Representations

The representations of S6(2) available are:
• Permutation representations, including all primitive ones.
• Permutations on 28 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 36 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 56 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - imprimitive.
• Permutations on 63 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 72 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - imprimitive.
• Permutations on 120 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 126 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - imprimitive.
• Permutations on 135 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 240 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - imprimitive.
• Permutations on 288 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - imprimitive.
• Permutations on 315 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 336 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 378 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - imprimitive, on the cosets of 2 × 2^4:S5.
• Permutations on 378 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - imprimitive, on the cosets of 2^5:PGL2(5).
• Permutations on 960 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 2.
• Dimension 6 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the natural representation.
• Dimension 8 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 14 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 48 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 64 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 112 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 512 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the Steinberg representation.
• All faithful irreducibles in characteristic 3.
• Dimension 7 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 14 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 21 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 27 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 34 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 35 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 49 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 91 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 98 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 189a over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - phi11 in the modular atlas.
• Dimension 189b over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - phi12 in the modular atlas.
• Dimension 189c over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - phi13 in the modular atlas.
• Dimension 196 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 405 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 0.
• Dimension 7 over Z: a and b; a and b; a and b (all Magma). - lattices E7*, E7, E7*.
• Dimension 7 over Q: a and b (Magma). - orthogonal (and over Z[½]).
• Dimension 15 over Z: a and b (Magma).
• Dimension 27 over Z: a and b (Magma).
• Dimension 35b over Z: a and b (Magma). - the deleted permutation representation.
The representations of 2.S6(2) available are:
• Some faithful permutation representations.
• Permutations on 240 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - with character (1 + 35a + 84) + (120[b]).
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.
• Permutations on 240 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - with character (1 + 35a + 84) + (8 + 112b).
• Permutations on 480 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - on the cosets of U3(3).
• Permutations on 1920 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - on the cosets of L2(8):3.
• Permutations on 2160 points: A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - on the cosets of 2^3.L3(2).
• All faithful irreducibles in characteristic 3.
• Dimension 8 over Z: A and B (Magma).

### Maximal subgroups

The maximal subgroups of S6(2) are as follows.

### Conjugacy classes

Representatives of the 30 conjugacy classes of S6(2) are given below.
• 1A: identity or a2.
• 2A: a.
• 2B: (ab2)6.
• 2C: [a, b2].
• 2D: (abab3ab6)3.
• 3A: (ab3)5.
• 3B: (ab)3.
• 3C: (abab3ab6)2.
• 4A: (ab2)3.
• 4B: (abab5)3.
• 4C: ababab5.
• 4D: (abab3)2.
• 4E: ababab2abab4.
• 5A: (ab3)3.
• 6A: ab2ab2ab3.
• 6B: abababab4.
• 6C: (ab2)2.
• 6D: (abab5)2.
• 6E: ababab2ab6.
• 6F: ab2ab2ab2ab6
• 6G: abab3ab6.
• 7A: b.
• 8A: abab4.
• 8B: abab3.
• 9A: ab.
• 10A: ababab5ab6.
• 12A: abab5.
• 12B: ababab6.
• 12C: ab2.
• 15A: ab3.
These are available here, and generators for the maximal cyclic subgroups are available here Go to main ATLAS (version 2.0) page. Go to classical groups page. Go to old S6(2) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 created on 7th June 2000.
Last updated 16.11.04 by JNB.
Information checked to Level 0 on 07.06.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.