# ATLAS: Linear group L4(3)

Order = 6065280 = 27.36.5.13.
Mult = 2.
Out = 2 × 2.

### Standard generators

Standard generators of L4(3) are a and b where a is in class 2A, b is in class 4B, ab has order 13 and abb has order 8. The last condition is equivalent to ababb has order 13. These conditions ensure that ab is in class 13A/B.
Standard generators of the double cover 2.L4(3) are preimages A and B where AB has order 13 and ABABB has order 13.

Standard generators of L4(3).21 are c and d where c is in class 2C, d has order 5, cd has order 26 and cdcdd has order 5.
Standard generators of the double cover 2.L4(3).21 are preimages C and D where D has order 5.

### Black box algorithms

To find standard generators for L4(3):
• Find x in class 2A - this may be done by taking a suitable power of an element of order 10 or 20.
• Find y in class 4B, for example as the square of an element of order 8.
• Find conjugates a of x and b of y such that ab has order 13 and abb has order 8.

### Automorphisms

The 21 automorphism may be realised by mapping (a,b) to (a,(abb)-1b(abb))
The 22 automorphism may be realised by mapping (a,b) to ((ab)-5a(ab)5, (abb)-2b(abb)2)
The 23 automorphism may be realised by mapping (a,b) to ((ab)-1b, (abb)-4b(abb)4)

### Representations

The representations of L4(3) available are
• a and b as permutations on 40 points.
• a and b as permutations on 40 points - the image of the above under certain outer automorphisms.
• a and b as permutations on 117 points.
• a and b as permutations on 117 points - the image of the above under certain outer automorphisms.
• a and b as permutations on 130 points.
• a and b as permutations on 2106 points.
• a and b as 26 × 26 matrices over GF(2).
• a and b as 26 × 26 matrices over GF(2).
• a and b as 38 × 38 matrices over GF(2).
• a and b as 6 × 6 matrices over GF(3) - the natural representation as O6+(3).
• a and b as 10 × 10 matrices over GF(3).
• a and b as 26 × 26 matrices over GF(5).
• a and b as 26 × 26 matrices over GF(5).
• a and b as 38 × 38 matrices over GF(5).
• a and b as 90 × 90 matrices over GF(5).
• a and b as 26 × 26 matrices over GF(13).
• a and b as 26 × 26 matrices over GF(13).
• a and b as 39 × 39 matrices over GF(13).
• a and b as 89 × 89 matrices over GF(13).
The representations of 2.L4(3) = SL4(3) available are
• A and B as permutations on 80 points.
• A and B as 4 × 4 matrices over GF(3) - the natural representation as SL4(3).
• A and B as 4 × 4 matrices over GF(3) - the dual of the above.
The representations of L4(3):2a available are
• c and d as 6 × 6 matrices over GF(3).
The representations of 2.L4(3):2a available are
• C and D as 4 × 4 matrices over GF(3).