ATLAS: Linear group L_{6}(2)
Order = 20158709760 = 2^{15}.3^{4}.5.7^{2}.31.
Mult = 1.
Out = 2.
Standard generators
Standard generators of L_{6}(2) are a
and b where
a is in class 2A, b is in class 6F,
ab has order 63 and abb has order 6.
Standard generators of L_{6}(2):2 are c
and d where
c is in class 2D,
d is in class 7CD (i.e. the class of 7elements with fixed points
in the natural representation)
cd has order 30, and
cdd has order 14.
To obtain standard generators for L6(2) from those for L6(2):2 run
this program.
The outer automorphism of L6(2) may be realised by running
this program.
Black box algorithms
To obtain standard generators of L6(2):
 Find an element of order 10 or 30. It powers to an element x in class 2A.
 Find a random element y of order 6. It is most likely to be in class 6F.
 Conjugate x and/or y at random until the product has order 63.
[This should happen about 5% of the time.]
 If you fail to get order 63, you will suspect that
y is not in class 6F, so go back two steps.
 If xy has order 63, then y is in class 6F.
 If xyy has order 21, continue conjugating x and/or y until both: xy has order 63,
and xyy has order 6.
To obtain standard generators of L6(2).2:
 Find an element of order 18. Its 9th power is an element x in class 2D.
 Find an element of order 28. Its 4th power is an element y in class 7CD.
 Conjugate x and/or y at random until the product has order 30.
 If xyy has order 30, go back to previous step.
 Otherwise xyy has order 14, and you have finished.
Representations
The representations of L_{6}(2) available are:

Permutations on 63 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Some irreducibles over GF(2):

Dimension 6 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.

Dimension 6 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 20 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 34 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 70 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 84 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 90 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 154 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 204 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 384 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 400 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 720 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 896 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 924 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 61 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 527 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 651 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 62 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 61 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 62 over GF(31):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of L_{6}(2):2 available are:

Permutations on 126 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Some irreducibles over GF(2) (all of degree up to 1000):

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

Dimension 20 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 30 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 34 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 140 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 154 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 168 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 180 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 400 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 408 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 768 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 61 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 61 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
Maximal subgroups
The maximal subgroups of L_{6}(2) are:
 2^{5}:L_{5}(2), the point stabiliser.
Order: 319979520.
Index: 63.
 2^{5}:L_{5}(2), the 4space stabiliser.
Order: 319979520.
Index: 63.
 2^{8}:(A_{8} × S_{3}), the line stabiliser.
Order: 30965760.
Index: 651.
 2^{8}:(A_{8} × S_{3}), the 3space stabiliser.
Order: 30965760.
Index: 651.
 2^{9}:(L_{3}(2) × L_{3}(2)), the plane stabiliser.
Order: 14450688.
Index: 1395.
 S_{6}(2).
Order: 1451520.
Index: 13888.
 3.L_{3}(4):S_{3}.
Order: 362880.
Index: 55552.
 (L_{3}(2) × L_{3}(2)):2.
Order: 56448.
Index: 357120.
 (L_{2}(8) × 7):3.
Order: 10584.
Index: 1904640.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to old L6(2) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 created on 14th December 2001.
Last updated 04.11.02 by RAW.
Information checked to
Level 0 on 05.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.