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: The representations of 2.L2(19):2 (NB not the Atlas group) available are:

Maximal subgroups

The maximal subgroups of L2(19) are as follows. The maximal subgroups of L2(19):2 are as follows.

Conjugacy classes

A set of generators for the maximal cyclic subgroups of L2(19) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

A set of generators for the maximal cyclic subgroups of L2(19):2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.


Main ATLAS page Go to main ATLAS (version 2.0) page.
Linear groups page Go to linear groups page.
Old L2(19) page Go to old L2(19) page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 file created on 18th January 2002, from Version 1 file last modified on 22.12.98.
Last updated 28.01.02 by RAW.
Information checked to Level 0 on 18.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.