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):

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 The representations of 2.L4(3) = SL4(3) available are The representations of L4(3):2a available are The representations of 2.L4(3):2a available are
Main ATLAS page Go to main ATLAS (version 2.0) page.
Linear groups page Go to linear groups page.
Old L4(3) page Go to old L4(3) page - ATLAS version 1.
ftp access 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.