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):

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
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Last updated 26th March 1999,
R.A.Wilson, R.A.Parker and J.N.Bray