ATLAS: Linear group L_{2}(8), Derived Ree group R(3)'
Order = 504 = 2^{3}.3^{2}.7.
Mult = 1.
Out = 3.
The following information is available for L_{2}(8):
Standard generators of L_{2}(8) are a and b where
a has order 2, b has order 3 and ab has order 7.
Standard generators of L_{2}(8):3 are c and d where
c has order 2, d has order 3, cd has order 9 and cdcdcd^{1}cdcd^{1}cd^{1} has order 7. The last condition is equivalent to: cdcdcd^{1}cd^{1}cdcd^{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 L_{2}(8).
To find standard generators for L_{2}(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 L_{2}(8).
To find standard generators for L_{2}(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^{1}cdcd^{1}cd^{1} has order 2 then invert d.
 The elements c and d are now standard generators of
L_{2}(8):3.
An outer automorphism of L_{2}(8) of order 3 may be obtained by mapping (a, b) to (a, b^{ababba}).
To obtain our standard generators for L_{2}(8):3 we may take c = babb and d to be the above automorphism.
Conversely, we may take a = cd^{1}cdcd^{1}cd^{1}cdcdcd^{1}cdc and b = cdcdcd^{1}cd^{1}cd^{1}cdcd^{1}cd. Note also that a' = c and b' = d^{1}(cd)^{3}d are equivalent under an automorphism to (a, b).
Presentations for L_{2}(8) and L_{2}(8):3 = R(3) in terms of their standard generators are given below.
< a, b  a^{2} = b^{3} = (ab)^{7} = (ababab^{1}ababab^{1}ab^{1})^{2} = 1 >.
< c, d  c^{2} = d^{3} = (cd)^{9} = [c, d]^{9} = (cdcdcd^{1}cd^{1}cdcd^{1})^{2} = 1 >.
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 L_{2}(8) available are:
 All primitive permutation representations.

Permutations on 9 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 28 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 36 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 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.

Dimension 7 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(3)  reducible over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 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 L_{2}(8):3 available are:
 All faithful primitive permutation representations.

Permutations on 9 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 28 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 36 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducible representations in characteristic 2 whose character appears in the ABC.

Dimension 6 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 8 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 12 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 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.

Dimension 7 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 8 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 21 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 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).
The maximal subgroups of L_{2}(8) are as follows.
The maximal subgroups of L_{2}(8):3 are as follows.
 L_{2}(8).
 2^{3}:7:3 = 2^{3}:F_{21}.
 9:6._{ }
 7:6 = F_{42}.
Representatives of the 9 conjugacy classes L_{2}(8) are given below.
 1A: identity [or a^{2}].
 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 L_{2}(8):3 are given below.
 1A: identity [or c^{2}].
 2A: c.^{ }
 3A: (cd)^{3}.
 7ABC: cdcdcd^{1}cd^{1} or [c, dcd].
 9ABC: cdcd^{1} or [c, d].
 3B: d.^{ }
 6A: cdcdcd^{1}.
 9D: cd.^{ }
 3B': d^{1} or d^{2}.
 6A': cdcd^{1}cd^{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.