# ATLAS: Linear group L3(3)

Order = 5616 = 24.33.13.
Mult = 1.
Out = 2.

### Standard generators

Standard generators of L3(3) are a and b where a has order 2, b is in class 3B and ab is in class 13A/B. The last condition is equivalent to: ab has order 13 and ababb has order 4.
Wlog define 13A to be the class containing ab, and then 8B is the class containing abababb.

Standard generators of L3(3):2 are c and d where c is in class 2B, d is in class 4B and cd is in class 13AB. The last condition is equivalent to: cd has order 13 and cdcdcdd has order 12.

### Representations

The representations of L3(3) available are
• All primitive permutation representations.
• Permutations on 13 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 13 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the image of the above under an outer automorphism.
• Permutations on 144 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Permutations on 234 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 2 (up to Frobenius automorphisms).
• All faithful irreducibles in characteristic 3.
• Dimension 3 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the natural representation.
• Dimension 3 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the dual and skew-square of the above.
• Dimension 6 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the symmetric square of the natural representation.
• Dimension 6 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the dual of the above.
• Dimension 7 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 15 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 15b 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). - the Steinberg representation.
• Faithful irreducibles in characteristic 13.
• Faithful irreducibles in characteristic 0.
• a and b as 12 × 12 matrices over Z.
• a and b as 13 × 13 monomial matrices over Z.
• a and b as 26 × 26 matrices over Z.
• a and b as 26 × 26 matrices over Z[i2].
• a and b as 26 × 26 matrices over Z[i2] - the dual of the above.
• a and b as 27 × 27 matrices over Z.
• a and b as 39 × 39 monomial matrices over Z.
• a and b as 52 × 52 matrices over Z - reducible over Q(i2).
• a and b as 64 × 64 matrices over Z - reducible over Q(b13) and Q(d13).
The representations of L3(3):2 available are
• Faithful permutation representations, including all primitive ones.
• Permutations on 26 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - imprimitive.
• Permutations on 52 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - primitive.
• Permutations on 117 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - primitive.
• Permutations on 144 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - primitive.
• Permutations on 234 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - primitive.
• All faithful irreducibles in characteristic 2.
• All faithful irreducibles in characteristic 3 (up to tensoring with linear characters).
• All faithful irreducibles in characteristic 13 (up to tensoring with linear characters).

### Maximal subgroups

The maximal subgroups of L3(3) are as follows.
• 3^2:2S4.
• 3^2:2S4.
• 13:3.
• S4.
The maximal subgroups of L3(3):2 are as follows.
• L3(3).
• 3^1+2.D8.
• 2.S4.2.
• 13:6.
• S4 × 2. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L3(3) page - ATLAS version 1. Anonymous ftp access is also available on for.mat.bham.ac.uk.

Version 2.0 created on 13th December 2001.
Last updated 13.12.01 by RAW.
Information checked to Level 0 on 13.12.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.