# ATLAS: Linear group L2(17)

Order = 2448 = 24.32.17.
Mult = 2.
Out = 2.

The following information is available for L2(17):

### Standard generators

Standard generators of L2(17) are a and b where a has order 2, b has order 3 and ab has order 17.
Standard generators of the double cover 2.L2(17) = SL2(17) are preimages A and B where B has order 3 and AB has order 17.

Standard generators of L2(17):2 = PGL2(17) are c and d where c is in class 2B, d has order 3, cd has order 16 and cdcdd has order 4. These conditions ensure that cd is in class 16B.
Standard generators of either of the double covers 2.L2(17).2 = 2.PGL2(17) are preimages C and D where D has order 3.

### Presentations

Presentations for L2(17) and L2(17):2 = PGL2(17) in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)17 = ((ab)5(ab-1)3)2 = 1 >.

< c, d | c2 = d3 = (cd)16 = [c, d]4 = [a, (ab)5]2 = 1 >.

### Representations

The representations of L2(17) available are:
• Permutations on 18 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 2 and over GF(2).
• All faithful irreducibles in characteristic 3.
• All faithful irreducibles in characteristic 17.
• Dimension 3 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - the natural representation as O3(17).
• Dimension 5 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 7 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 9 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 11 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 13 over GF(17): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• 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).
The representations of 2.L2(17) = SL2(17) available are:
• Dimension 8 over GF(9): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 8 over GF(9): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• All faithful irreducibles in characteristic 17.
The representations of L2(17):2 = PGL2(17) available are:
The representations of 2L2(17).2 available are:

### Maximal subgroups

The maximal subgroups of L2(17) are as follows.
• F136 = 17:8.
• S4.
• S4.
• D18.
• D16.
The maximal subgroups of L2(17):2 = PGL2(17) are as follows.
• L2(17).
• F272 = 17:16.
• D36.
• D32.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of L2(17) 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 11 conjugacy classes of L2(17) are as follows.

• 1A: identity.
• 2A: a.
• 3A: b.
• 4A: [a, bab].
• 8A: (ab)3(ab-1)2.
• 8B: (ab)3ab-1.
• 9A: [a, b].
• 9B: [a, b]2.
• 9C: ababab-1 or [a, b]4.
• 17A: ab.
• 17B: (ab)3.
The 19 conjugacy classes of L2(17):2 = PGL2(17) are as follows.
• 1A: identity.
• 2A: [c, d]2.
• 3A: d.
• 4A: [c, d].
• 8A: (cd)6.
• 8B: (cd)2.
• 9A: (cd)5cd-1.
• 9B: [c, dcdcd].
• 9C: [c, dcd].
• 17AB: (cd)6(cd-1)2.
• 2B: c.
• 6A: (cd)4cd-1.
• 16A: (cd)5.
• 16B: cd.
• 16C: (cd)3.
• 16D: (cd)7.
• 18A: (cd)7(cd-1)2.
• 18B: (cd)3(cd-1)2.
• 18C: cdcdcd-1. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L2(17) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 file created on 11th April 2000.
Last updated 21.01.02 by RAW.
Information checked to Level 0 on 18.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.