ATLAS: Linear group L_{2}(11)
Order = 660 = 2^{2}.3.5.11.
Mult = 2.
Out = 2.
Standard generators
Standard generators of L_{2}(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.L_{2}(11) =
SL_{2}(11) are preimages A and B
where B has order 3 and AB has order 11.
Standard generators of L_{2}(11):2 = PGL_{2}(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.L_{2}(11).2 = 2.PGL_{2}(11) are preimages C and D where
D has order 3.
Presentations
Presentations for L_{2}(11) and L_{2}(11):2 = PGL_{2}(11) in terms of their standard generators are given below.
< a, b | a^{2} = b^{3} = (ab)^{11} = [a, babab]^{2} = 1 >.
< c, d | c^{2} = d^{3} = (cd)^{10} = (cdcdcd^{-1})^{4} = [c, dcdcd^{-1}cd^{-1}cd^{-1}cdcd] = 1 >.
Representations
The representations L_{2}(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.L_{2}(11) = SL_{2}(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 L_{2}(11):2 = PGL_{2}(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 L_{2}(11) are as follows.
The maximal subgroups of L_{2}(11):2 = PGL_{2}(11) are as follows.
- L_{2}(11), with standard generators (cdcdcdd)^2, d.
- F_{110} = 11:10, with generators c, dcdcdcdcddcdd.
- S_{4}, with generators c, ddcdcd.
- D_{24}, with generators c, (cdcddcd)^2.
- D_{20}, with generators c, (dcdcdcd)^2.
The programs for the maximal subgroups of L_{2}(11) and PGL_{2}(11) are not yet available.
Conjugacy classes
The 8 conjugacy classes of L_{2}(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 L_{2}(11):2 = PGL_{2}(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^{-1}cdcd^{-1}.
- 12B: cdcdcdcd^{-1}cd^{-1}.
Return to main ATLAS page.
Last updated 29th March 1999,
R.A.Wilson, R.A.Parker and J.N.Bray