# ATLAS: Linear group L3(13)

Order = 270178272 = 25.32.7.133.61.
Mult = 3.
Out = S3.

The following information is available for L3(13):

### Standard generators

Standard generators of L3(13) are a and b where a has order 2, b has order 3, ab has order 61 and ababb has order 4. (Extra condition added 1/11/05.)
Standard generators of the triple cover 3.L3(13) = SL3(13) are not defined.

Standard generators of L3(13):2 are not defined.
Standard generators of L3(13):3 = PGL3(13) are not defined.
Standard generators of L3(13):S3 are not defined.

### Black box algorithms

To find standard generators of L3(13):
• Find an element of even order. This powers up to a of order 2.
• Find an element of order divisible by 3. This powers up to t of order 3.
• Find a conjugate b of t such that ab has order 61 and ababb has order 4.
• The elements a and b are standard generators.

### Representations

The representations of L3(13) available are:
• Permutations on 183a points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). (Corrected 1/11/05)
• Permutations on 183b points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). (Corrected 1/11/05)
• Dimension 182 over Z: a and b (GAP).
• Dimension 183 over Z: a and b (GAP).
• Dimension 183b over Z[i]: a and b (GAP).
• Dimension 183c over Z[i]: a and b (GAP).
• Dimension 366 over Z: a and b (GAP). — reducible over Q(i).
The representations of 3.L3(13) = SL3(13) available are:
• none

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Version 2.0 file created on 3rd August 2004.
Last updated 13.12.05 by JNB.