# ATLAS: Linear group L2(8), Derived Ree group R(3)'

Order = 504 = 23.32.7.
Mult = 1.
Out = 3.

The following information is available for L2(8):

### Standard generators

Standard generators of L2(8) are a and b where a has order 2, b has order 3 and ab has order 7.

Standard generators of L2(8):3 are c and d where c has order 2, d has order 3, cd has order 9 and cdcdcd-1cdcd-1cd-1 has order 7. The last condition is equivalent to: cdcdcd-1cd-1cdcd-1 has order 2. Note that these conditions imply that d is conjugate to the field automorphism that squares the matrix entries in the natural representation of L2(8).

### Black box algorithms

To find standard generators for L2(8):
• Find any element x of order 2.
[The probability of success at each attempt is 1 in 8.]
• Find any element of order 3 or 9. This powers to y of order 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 whose product has order 7.
[The probability of success at each attempt is 3 in 7 (about 1 in 2).]
• The elements a and b are now standard generators of L2(8).
To find standard generators for L2(8).3:
• Find any element of order 6. This cubes to x of order 2 and squares to y of order 3.
[The probability of success at each attempt is 1 in 3.]
• Find a conjugate c of x and a conjugate d of y whose product has order 9.
[The probability of success at each attempt is 2 in 7 (about 1 in 4).]
• If cdcdcd-1cdcd-1cd-1 has order 2 then invert d.
• The elements c and d are now standard generators of L2(8):3.

### Automorphisms

An outer automorphism of L2(8) of order 3 may be obtained by mapping (a, b) to (a, bababba).

To obtain our standard generators for L2(8):3 we may take c = babb and d to be the above automorphism.
Conversely, we may take a = cd-1cdcd-1cd-1cdcdcd-1cdc and b = cdcdcd-1cd-1cd-1cdcd-1cd. Note also that a' = c and b' = d-1(cd)3d are equivalent under an automorphism to (a, b).

### Presentations

Presentations for L2(8) and L2(8):3 = R(3) in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)7 = (ababab-1ababab-1ab-1)2 = 1 >.

< c, d | c2 = d3 = (cd)9 = [c, d]9 = (cdcdcd-1cd-1cdcd-1)2 = 1 >.

### Representations

It was intended that these representations be ordered with respect to the class labellings given below, but please check this yourself if you rely on it.

The representations of L2(8) available are:

• All primitive permutation representations.
• All faithful irreducibles in characteristic 2.
• Dimension 2 over GF(8) - the natural representation: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 2 over GF(8): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 2 over GF(8): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 4 over GF(8): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 4 over GF(8): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 4 over GF(8): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 6 over GF(2) - reducible over GF(8): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 8 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 12 over GF(2) - reducible over GF(8): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All faithful irreducibles in characteristic 3.
• Essentially all faithful irreducibles in characteristic 7.
• Dimension 7 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 8 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 21 over GF(7) - reducible over GF(343): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Some faithful irreducibles in characteristic 0.
• Dimension 8 over Z: a and b (Magma).
• Dimension 27 over Z - reducible over Q(y7): a and b (Magma).
The representations of L2(8):3 available are:
• All faithful primitive permutation representations.
• All faithful irreducible representations in characteristic 2 whose character appears in the ABC.
• Both faithful irreducible representations in characteristic 3.
• Dimension 7 over GF(3) - the natural representation as R(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 27 over GF(3): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• All faithful irreducible representations in characteristic 7 whose character appears in the ABC.
• All faithful irreducible representations in characteristic 0 whose character appears in the ATLAS.
• Dimension 7 over Z: c and d (Magma).
• Dimension 8 over Z: c and d (Magma).
• Dimension 21 over Z: c and d (Magma).
• Dimension 27 over Z: c and d (Magma).

### Maximal subgroups

The maximal subgroups of L2(8) are as follows.
The maximal subgroups of L2(8):3 are as follows.
• L2(8).
• 23:7:3 = 23:F21.
• 9:6.
• 7:6 = F42.

### Conjugacy classes

Representatives of the 9 conjugacy classes L2(8) are given below.
• 1A: identity [or a2].
• 2A: a.
• 3A: b.
• 7A: ab.
• 7B: (ab)2.
• 7C: (ab)3 or (ab)4.
• 9A: abab-1 or [a, b].
• 9B: [a, b]2.
• 9C: [a, bab] or [a, b]4.
A program to calculate them is given here and a program to calculate representatives of the maximal cyclic subgroups is given here.

Representatives of the 11 conjugacy classes L2(8):3 are given below.

• 1A: identity [or c2].
• 2A: c.
• 3A: (cd)3.
• 7ABC: cdcdcd-1cd-1 or [c, dcd].
• 9ABC: cdcd-1 or [c, d].
• 3B: d.
• 6A: cdcdcd-1.
• 9D: cd.
• 3B': d-1 or d2.
• 6A': cdcd-1cd-1.
• 9D': cd-1.
A program to calculate them is given here and a program to calculate representatives of the maximal cyclic subgroups is given here. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L2(8) page - ATLAS version 1. Anonymous ftp access is also available on sylow.mat.bham.ac.uk.

Version 2.0 created on 15th April 1999.
Last updated 15.04.99 by JNB.
Information checked to Level 1 on 15.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.