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): To find 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: The representations of 2.L2(19) = SL2(19) available are: The representations of L2(19):2 = PGL2(19) available are:
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Last updated 22nd December 1998,
R.A.Wilson, R.A.Parker and J.N.Bray