ATLAS: Linear group L2(19)

Order = 3420 = 22.32.5.19.
Mult = 2.
Out = 2.

The following information is available for L2(19):

Standard generators

Standard generators of L2(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.L2(19) = SL2(19) are preimages A and B where B has order 3 and AB has order 19.

Standard generators of L2(19):2 = PGL2(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.L2(19).2 = 2.PGL2(19) are preimages C and D where D has order 3.

Automorphisms

An outer automorphism of L2(19) of order 2 may be obtained by mapping (a, b) to (a, b-1).

Black box algorithms

To find standard generators of L2(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 L2(19).
To find standard generators of L2(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 L2(19):2.

Presentations

Presentations of L2(19) and L2(19):2 = PGL2(19) on their standard generators are given below.

< a, b | a2 = b3 = (ababab-1)5 = [a, bab(ab-1)3abab] = 1 >.

< c, d | c2 = d3 = (cd)20 = [c, d]5 = ((cd)4(cd-1)3)2 = 1 >.

Representations

The representations of L2(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.L2(19) = SL2(19) available are:
• A and B as permutations on 40 points.
• A and B as 2 × 2 matrices over GF(19).
The representations of L2(19):2 = PGL2(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).