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 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 L_{6}(2) available are:
- a and
b as
permutations on 63 points.
- a and
b as
6 × 6 matrices over GF(2) - the natural representation.
- a and
b as
61 × 61 matrices over GF(3).
- a and
b as
61 × 61 matrices over GF(7).
The representations of L_{6}(2):2 available are:
- c and
d as
12 × 12 matrices over GF(2).
- c and
d as
61 × 61 matrices over GF(3).
- c and
d as
61 × 61 matrices over GF(7).
- c and
d as
permutations on 126 points.
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 4-space 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 3-space 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.
Return to main ATLAS page.
Last updated 14th August 1998,
R.A.Wilson, R.A.Parker and J.N.Bray