# 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.

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.
• a and b as permutations on 13 points.
• a and b as permutations on 13 points - the image of the above under an outer automorphism.
• a and b as permutations on 144 points.
• a and b as permutations on 234 points.
• Faithful irreducibles in characteristic 2.
• a and b as 12 × 12 matrices over GF(2).
• a and b as 16 × 16 matrices over GF(16).
• a and b as 26 × 26 matrices over GF(2).
• All faithful irreducibles in characteristic 3.
• a and b as 3 × 3 matrices over GF(3) - the natural representation.
• a and b as 3 × 3 matrices over GF(3) - the dual and skew-square of the above.
• a and b as 6 × 6 matrices over GF(3) - the symmetric square of the natural representation.
• a and b as 6 × 6 matrices over GF(3) - the dual of the above.
• a and b as 7 × 7 matrices over GF(3).
• a and b as 15 × 15 matrices over GF(3).
• a and b as 15 × 15 matrices over GF(3) - the dual of the above.
• a and b as 27 × 27 matrices over GF(3) - the Steinberg representation.
• Faithful irreducibles in characteristic 13.
• a and b as 11 × 11 matrices over GF(13).
• a and b as 13 × 13 monomial matrices over GF(13).
• a and b as 16 × 16 matrices over GF(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.
• c and d as permutations on 26 points - imprimitive.
• c and d as permutations on 52 points - primitive.
• c and d as permutations on 117 points - primitive.
• c and d as permutations on 144 points - primitive.
• c and d as permutations on 234 points - primitive.
• c and d as 12 × 12 matrices over GF(2).
• c and d as 6 × 6 matrices over GF(3).
• c and d as 11 × 11 matrices over GF(13).
• c and d as 13 × 13 matrices over GF(13).

### 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.