# 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.
• All irreducibles in characteristic 2:
• All irreducibles in characteristic 3:
• All irreducibles in characteristic 5:
• All irreducibles over GF(19):
The representations of 2.L2(19) = SL2(19) available are:
The representations of L2(19):2 = PGL2(19) available are:
• Permutation representations, including all faithful primitive ones.
• Dimension 3 over GF(19): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
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.
• 19:9, with generators ???.
• A5
• A5
• D20
• D18
The maximal subgroups of L2(19):2 are as follows.
• L2(19)
• 19:18
• D40
• D36
• S4

### 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. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L2(19) page - ATLAS version 1. 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.