# ATLAS: Linear group L2(3)

Order = 1092.
Mult = 2.
Out = 2.

The following information is available for L2(13):

### Standard generators

Standard generators of L2(13) are a and b where a has order 2, b has order 3 and ab has order 13.
Standard generators of the double cover 2.L2(13) = SL2(13) are pre-images A and B where B has order 3 and AB has order 13.

Standard generators of L2(13).2 are c and d where c has order 2 (necessarily class 2B), d has order 4 and cd has order 13.
Standard generators of the double cover 2.L2(13).2 are pre-images C and D where CD has order 13.

### Automorphisms

The outer automorphism of L2(13) may be achieved by applying this program to the standard generators.

### Representations

The representations available are
• Permutations on 14 points: a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All irreducibles in characteristic 2:
• Dimension 13 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 12 over GF(7): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• All irreducibles in characteristic 13:
The representations of 2.L2(13) available are
• All irreducibles in characteristic 3:
• All irreducibles in characteristic 7:
• All irreducibles in characteristic 13:
• Dimension 2 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP). - the natural representation.
• Dimension 4 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 6 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 8 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 10 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
• Dimension 12 over GF(13): A and B (Meataxe), A and B (Meataxe binary), A and B (GAP).
The representations of L2(13).2 available are
The representations of 2L2(13).2 available are

### Maximal subgroups

The maximal subgroups of L2(13) are as follows.
• 13:6
• D14
• D12
• A4
The maximal subgroups of L2(13):2 are as follows.
• L2(13).
• D28.
• D24
• S4

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of L2(13) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.
In the double cover, the element ABABABB is in class +6A, for compatibility with the ABC.

A set of generators for the maximal cyclic subgroups of L2(13):2 can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

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Go to old L2(13) page - ATLAS version 1.
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Version 2.0 file created on 21st January 2002, from Version 1 file last updated on 13.03.96.
Last updated 29.01.02 by RAW.
Information checked to Level 0 on 21.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.