# ATLAS: Linear group L2(11)

Order = 660 = 22.3.5.11.
Mult = 2.
Out = 2.

The following information is available for L2(11):

### Standard generators

Standard generators of L2(11) are a and b where a has order 2, b has order 3 and ab has order 11.
Standard generators of the double cover 2.L2(11) = SL2(11) are preimages A and B where B has order 3 and AB has order 11.

Standard generators of L2(11):2 = PGL2(11) are c and d where c is in class 2B, d has order 3, cd has order 10 and cdcdd has order 11. These conditions imply that cd is in class 10A.
Standard generators of either double cover 2.L2(11).2 = 2.PGL2(11) are preimages C and D where D has order 3.

### Presentations

Presentations for L2(11) and L2(11):2 = PGL2(11) in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)11 = [a, babab]2 = 1 >.

< c, d | c2 = d3 = (cd)10 = (cdcdcd−1)4 = [c, dcdcd−1cd−1cd−1cdcd] = 1 >.

We have < c, d | c2 = d3 = (cd)10 = (cdcdcd−1)4 = 1 > = (3 × L2(11)):2 (with trivial centre). To quotient out the normal 3, we can add a relation such as [c, d]11 = 1, though the shortest such relation (when written in terms of c, d, c−1, d−1) is [c, dcdcd−1cd−1cd−1cdcd] = 1, which we have used above.

### Representations

The representations L2(11) available are:
• All primitive permutation representations.
• All faithful irreducibles in characteristic 2.
• Dimension 5 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 5 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 12 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 12 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 10 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(4).
• Dimension 24 over GF(2): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP). - reducible over GF(4).
• All faithful irreducibles in characteristic 3.
• All faithful irreducibles in characteristic 5.
• All faithful irreducibles in characteristic 11.
• Some faithful irreducibles in characteristic 0 and over Z.
• a and b as 5 × 5 matrices over Z[b11].
• a and b as 5 × 5 matrices over Z[b11] - the dual of the above.
• a and b as 10 × 10 matrices over Z.
• a and b as 10 × 10 matrices over Z.
• a and b as 11 × 11 matrices over Z.
The representations of 2.L2(11) = SL2(11) available are:
• All faithful irreducibles in characteristic 11.
The representations of L2(11):2 = PGL2(11) available are:
• Faithful permutation representations, including all primitive ones.
• Permutations on 12 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Permutations on 22 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - imprimitive.
• Permutations on 55 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - on the cosets of S4.
• Permutations on 55 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - on the cosets of D24.
• Permutations on 66 points: c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• All faithful irreducibles in characteristic 2 and over GF(2) (in ABC order).
• Dimension 10 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - restriction to L2(11) reducible over GF(4).
• Dimension 10 over GF(2): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP). - restriction to L2(11) absolutely irreducible.
• Dimension 12 over GF(4): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• Dimension 12 over GF(4): c and d (Meataxe), c and d (Meataxe binary), c and d (GAP).
• c and d as 24 × 24 matrices over GF(2) - reducible over GF(4).
• All faithful irreducibles in characteristic 11 (with character in ABC) up to tensoring with linear irredicibles.

### Maximal subgroups

The maximal subgroups of L2(11) are as follows.
The maximal subgroups of L2(11):2 = PGL2(11) are as follows.

### Conjugacy classes

A set of generators for the maximal cyclic subgroups of L2(11) can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.

The 8 conjugacy classes of L2(11) are as follows.

• 1A: identity.
• 2A: a.
• 3A: b.
• 5A: abababb.
• 5B: ababb or [a, b].
• 6A: ababababb or [a, bab].
• 11A: ab.
• 11B: abb.

A set of generators for the maximal cyclic subgroups of L2(11):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.

The 13 conjugacy classes of L2(11):2 = PGL2(11) are as follows.

• 1A: identity.
• 2A: (cdcdcd-1)2.
• 3A: d.
• 5A: (cd)4.
• 5B: (cd)2.
• 6A: cdcdcdcd-1 or [c, dcd].
• 11AB: [c, d].
• 2A: c.
• 4A: cdcdcd-1.
• 10A: cd.
• 10B: (cd)3.
• 12A: cdcdcd-1cdcd-1.
• 12B: cdcdcdcd-1cd-1. Go to main ATLAS (version 2.0) page. Go to linear groups page. Go to old L2(11) page - ATLAS version 1. Anonymous ftp access is also available. See here for details.

Version 2.0 file created on 21st January 2002, from Version 1 file last updated on 29.03.99.
Last updated 26.01.12 by JNB.
Information checked to Level 0 on 21.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.