ATLAS: Linear group L6(2)
Order = 20158709760 = 215.34.5.72.31.
Mult = 1.
Out = 2.
Standard generators
Standard generators of L6(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 L6(2):2 are c
and d where
c is in class 2D,
d is in class 7CD (i.e. the class of 7-elements 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 L6(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 L6(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 L6(2) are:
- 25:L5(2), the point stabiliser.
Order: 319979520.
Index: 63.
- 25:L5(2), the 4-space stabiliser.
Order: 319979520.
Index: 63.
- 28:(A8 × S3), the line stabiliser.
Order: 30965760.
Index: 651.
- 28:(A8 × S3), the 3-space stabiliser.
Order: 30965760.
Index: 651.
- 29:(L3(2) × L3(2)), the plane stabiliser.
Order: 14450688.
Index: 1395.
- S6(2).
Order: 1451520.
Index: 13888.
- 3.L3(4):S3.
Order: 362880.
Index: 55552.
- (L3(2) × L3(2)):2.
Order: 56448.
Index: 357120.
- (L2(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.