# ATLAS: Linear group L2(16)

Order = 4080 = 24.3.5.17.
Mult = 1.
Out = 4.

The following information is available for L2(16):

### Standard generators

Standard generators of L2(16) are a and b where a has order 2, b has order 3 and ab has order 15.

Standard generators of L2(16):2 are c and d where c is in class 2B, d has order 4 and cd has order 15.

Standard generators of L2(16):4 are e and f where e is in class 2A, f is in class 4B/B', ef has order 8, eff has order 6 and efeffefff has order 4. .

### Automorphisms

The outer automorphism of order 2 of L2(16) may be achieved by applying this program to the standard generators.
The outer automorphism of order 4 of L2(16) may be achieved by applying this program to the standard generators. (Modulo inner automorphisms, this is the Frobenius automorphism *2.)

### Presentations

Presentations for L2(16), L2(16):2 and L2(16):4 in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)15 = ((ab)5(ab-1)3)2 = 1 >.

< c, d | c2 = d4 = [c, dcd]2 = cd(cdcd2)2(cd2cd)2cdcd-1 = 1 >.

< e, f | e2 = f4 = (ef)8 = (ef2)6 = (ef)3(ef-1)3(ef)2(ef-1)2(ef)2ef2 = 1 >.

### Representations

The representations of L2(16) available are
• Permutations on 17 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 2 over GF(16): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the natural representation.
• Dimension 16 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 17 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 17 over GF(9): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 16 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles over GF(17):
• Dimension 15 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 17 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 34 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(289).
• Dimension 68 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - becomes dimension 17 over GF(17^4).
The representations of L2(16):2 available are
The representations of L2(16):4 available are

### Maximal subgroups

The maximal subgroups of L2(16) are as follows.
• 24:15.
• A5.
• D34.
• D30.
The maximal subgroups of L2(16):2 are as follows.
• L2(16).
• 24:(3 × D10).
• A5 × 2.
• F68 = 17:4.
• D10 × S3.
The maximal subgroups of L2(16):4 are as follows.
• L2(16):2.
• 24:15:4.
• (A5 × 2).2.
• F136 = 17:8.
• 5:4 × S3.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of L2(16) 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(16):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.

A set of generators for the maximal cyclic subgroups of L2(16):4 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

The classes of L2(16).4 are chosen so that efefff = [e,f] is in class 17EFGH. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L2(16) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 file created on 24th January 2002, from Version 1 file last updated on 08.10.98.
Last updated 29.01.02 by RAW.
Information checked to Level 0 on 24.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.