# 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:
The representations of L6(2):2 available are:

### 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.
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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.