ATLAS: Linear group L_{2}(19)
Order = 3420 = 2^{2}.3^{2}.5.19.
Mult = 2.
Out = 2.
The following information is available for L_{2}(19):
Standard generators
Standard generators of L_{2}(19) are a and b where
a has order 2, b has order 3 and ab has order 19.
Standard generators of the double cover 2.L_{2}(19) = SL_{2}(19)
are preimages A and B where B has order 3
and AB has order 19.
Standard generators of L_{2}(19):2 = PGL_{2}(19) are c
and d where c is in class 2B, d has order 3, cd has
order 20 and cdcdd has order 5.
Standard generators of either of the double covers 2.L_{2}(19).2 =
2.PGL_{2}(19) are preimages C and D where D has
order 3.
Automorphisms
An outer automorphism of L_{2}(19) of order 2 may be obtained by
mapping (a, b) to (a, b^{-1}).
Black box algorithms
To find standard generators of L_{2}(19):
- Find any element of order 2, x say, by taking a suitable power of any element of even order.
[The probability of success at each attempt is 1 in 4.]
- Find any element of order 3, y say, by taking a suitable power of any element of order divisible by 3.
[The probability of success at each attempt is 4 in 9 (about 1 in 2).]
- Find a conjugate a of x and a conjugate b of y such that ab has order 19.
[The probability of success at each attempt is 2 in 19 (about 1 in 10).]
- Now a and b are standard generators of L_{2}(19).
To find standard generators of L_{2}(19).2:
- Find any element of order 6 or 18. This powers up to x in class 2B.
[The probability of success at each attempt is 2 in 9 (about 1 in 5) OR 4 in 9 (about 1 in 2) if we restrict our search to outer elements only.]
- Find any element of order 3, y say, by taking a suitable power of any element of order divisible by 3.
[The probability of success at each attempt is 4 in 9 (about 1 in 2).]
- Find a conjugate c of x and a conjugate d of y such that cd has order 20 and cdcdd has order 5.
[The probability of success at each attempt is 9 in 95 (about 1 in 11).]
- Now c and d are standard generators of L_{2}(19):2.
Presentations
Presentations of L_{2}(19) and L_{2}(19):2 = PGL_{2}(19)
on their standard generators are given below.
< a, b | a^{2} = b^{3} = (ababab^{-1})^{5} = [a, bab(ab^{-1})^{3}abab] = 1 >.
< c, d | c^{2} = d^{3} = (cd)^{20} = [c, d]^{5} = ((cd)^{4}(cd^{-1})^{3})^{2} = 1 >.
Representations
The representations of L_{2}(19) available are:
- All primitive permutation representations.
- a and
b as
permutations on 20 points.
- a and
b as
permutations on 57 points.
- a and
b as
permutations on 57 points.
- a and
b as
permutations on 171 points.
- a and
b as
permutations on 190 points.
- a and
b as
3 × 3 matrices over GF(19).
- a and
b as
5 × 5 matrices over GF(19).
The representations of 2.L_{2}(19) = SL_{2}(19) available are:
- A and
B as
permutations on 40 points.
- A and
B as
2 × 2 matrices over GF(19).
The representations of L_{2}(19):2 = PGL_{2}(19) available are:
- Permutation representations, including all faithful primitive ones.
- c and
d as
permutations on 20 points.
- c and
d as
permutations on 114 points - imprimitive.
- c and
d as
permutations on 171 points.
- c and
d as
permutations on 190 points.
- c and
d as
permutations on 285 points.
- c and
d as
3 × 3 matrices over GF(19).
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Last updated 22nd December 1998,
R.A.Wilson, R.A.Parker and J.N.Bray