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): To find standard generators for 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:

The representations of L2(8):3 available are:

Maximal subgroups

The maximal subgroups of L2(8) are as follows. The maximal subgroups of L2(8):3 are as follows.

Conjugacy classes

Representatives of the 9 conjugacy classes L2(8) are given below. 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.

A program to calculate them is given here and a program to calculate representatives of the maximal cyclic subgroups is given here.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Linear groups page Go to linear groups page.
Old L2(8) page Go to old L2(8) page - ATLAS version 1.
ftp access 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.