# ATLAS: Linear group L2(11)

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

### 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 >.

### Representations

The representations L2(11) available are:
• All primitive permutation representations.
• a and b as permutations on 11 points.
• a and b as permutations on 11 points.
• a and b as permutations on 12 points.
• a and b as permutations on 55 points.
• All faithful irreducibles in characteristic 2 and over GF(2).
• a and b as 5 × 5 matrices over GF(4).
• a and b as 5 × 5 matrices over GF(4).
• a and b as 10 × 10 matrices over GF(2) - reducible over GF(4).
• a and b as 10 × 10 matrices over GF(2).
• a and b as 12 × 12 matrices over GF(4).
• a and b as 12 × 12 matrices over GF(4).
• a and b as 24 × 24 matrices over GF(2) - reducible over GF(4).
• All faithful irreducibles in characteristic 3 and over GF(3).
• a and b as 5 × 5 matrices over GF(3).
• a and b as 5 × 5 matrices over GF(3).
• a and b as 10 × 10 matrices over GF(3).
• a and b as 12 × 12 matrices over GF(9).
• a and b as 12 × 12 matrices over GF(9).
• a and b as 24 × 24 matrices over GF(3) - reducible over GF(9).
• All faithful irreducibles in characteristic 5.
• a and b as 5 × 5 matrices over GF(5).
• a and b as 5 × 5 matrices over GF(5).
• a and b as 10 × 10 matrices over GF(5).
• a and b as 10 × 10 matrices over GF(5).
• a and b as 11 × 11 matrices over GF(5).
• All faithful irreducibles in characteristic 11.
• a and b as 3 × 3 matrices over GF(11) - the natural representation as O3(11).
• a and b as 5 × 5 matrices over GF(11).
• a and b as 7 × 7 matrices over GF(11).
• a and b as 9 × 9 matrices over GF(11).
• a and b as 11 × 11 matrices over GF(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.
• A and B as 2 × 2 matrices over GF(11).
• A and B as 4 × 4 matrices over GF(11).
• A and B as 6 × 6 matrices over GF(11).
• A and B as 8 × 8 matrices over GF(11).
• A and B as 10 × 10 matrices over GF(11).
The representations of L2(11):2 = PGL2(11) available are:
• Faithful permutation representations, including all primitive ones.
• c and d as permutations on 12 points.
• c and d as permutations on 22 points - imprimitive.
• c and d as permutations on 55 points - on the cosets of S4.
• c and d as permutations on 55 points - on the cosets of D24.
• c and d as permutations on 66 points.
• All faithful irreducibles in characteristic 2 and over GF(2) (in ABC order).
• c and d as 10 × 10 matrices over GF(2) - restriction to L2(11) reducible over GF(4).
• c and d as 10 × 10 matrices over GF(2) - restriction to L2(11) absolutely irreducible.
• c and d as 12 × 12 matrices over GF(4).
• c and d as 12 × 12 matrices over GF(4).
• 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.
• c and d as 3 × 3 matrices over GF(11).
• c and d as 5 × 5 matrices over GF(11).
• c and d as 7 × 7 matrices over GF(11).
• c and d as 9 × 9 matrices over GF(11).
• c and d as 11 × 11 matrices over GF(11).

### Maximal subgroups

The maximal subgroups of L2(11) are as follows.
The maximal subgroups of L2(11):2 = PGL2(11) are as follows.
The programs for the maximal subgroups of L2(11) and PGL2(11) are not yet available.

### Conjugacy classes

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.
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.