ATLAS: Linear group L_{4}(3)
Order = 6065280 = 2^{7}.3^{6}.5.13.
Mult = 2.
Out = 2 × 2.
Standard generators
Standard generators of L_{4}(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.L_{4}(3) are
preimages A and B where
AB has order 13 and ABABB has order 13.
Standard generators of L_{4}(3).2_{1} 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.L_{4}(3).2_{1} are
preimages C and D where
D has order 5.
Black box algorithms
To find standard generators for L_{4}(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 2_{1} automorphism may be realised by mapping (a,b) to
(a,(abb)^{-1}b(abb))
The 2_{2} automorphism may be realised by mapping (a,b) to
((ab)^{-5}a(ab)^{5},
(abb)^{-2}b(abb)^{2})
The 2_{3} automorphism may be realised by mapping (a,b) to
((ab)^{-1}b,
(abb)^{-4}b(abb)^{4})
Representations
The representations of L_{4}(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.L_{4}(3) = SL_{4}(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 L_{4}(3):2a available are
- c and
d as
6 × 6 matrices over GF(3).
The representations of 2.L_{4}(3):2a available are
- C and
D as
4 × 4 matrices over GF(3).
Return to main ATLAS page
Last updated 26th March 1999,
R.A.Wilson, R.A.Parker and J.N.Bray