# ATLAS: Linear group L4(3)

Order = 6065280 = 27.36.5.13.
Mult = 2.
Out = 2 × 2.
See also ATLAS of Finite Groups, pp68-69.

### 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).
• Some faithful irreducibles in characteristic 0
• Dimension 26 over Z: a and b (GAP).
• Dimension 39 over Z: a and b (GAP).
• Dimension 52 over Z: a and b (GAP).
• Dimension 65 over Z: a and b (GAP).
• Dimension 90 over Z: a and b (GAP).
• Dimension 234 over Z: a and b (GAP).
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). Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L4(3) page - ATLAS version 1. Anonymous ftp access is also available on for.mat.bham.ac.uk, user atlasftp, password atlasftp. Files can be found in directory v2.0 and subdirectories.

Version 2.0 created on 4th March 2004
R.A.Wilson, R.A.Parker and J.N.Bray.