ATLAS: Linear group L_{2}(17)
Order = 2448 = 2^{4}.3^{2}.17.
Mult = 2.
Out = 2.
Standard generators
Standard generators of L_{2}(17) are a and b where
a has order 2, b has order 3 and ab has order 17.
Standard generators of the double cover 2.L_{2}(17) =
SL_{2}(17) are preimages A and B where
B has order 3 and AB has order 17.
Standard generators of L_{2}(17):2 = PGL_{2}(17) are
c and d where c is in class 2B, d has order 3,
cd has order 16 and cdcdd has order 4.
These conditions ensure that cd is in class 16B.
Standard generators of either of the double covers 2.L_{2}(17).2 =
2.PGL_{2}(17) are preimages C and D where
D has order 3.
Presentations
Presentations for L_{2}(17) and L_{2}(17):2 = PGL_{2}(17) in terms of their standard generators are given below.
< a, b | a^{2} = b^{3} = (ab)^{17} = ((ab)^{5}(ab^{-1})^{3})^{2} = 1 >.
< c, d | c^{2} = d^{3} = (cd)^{16} = [c, d]^{4} = [a, (ab)^{5}]^{2} = 1 >.
Representations
The representations of L_{2}(17) available are:
- a and
b as
permutations on 18 points.
- All faithful irreducibles in characteristic 2 and over GF(2).
- a and
b as
8 × 8 matrices over GF(2).
- a and
b as
8 × 8 matrices over GF(2).
- a and
b as
16 × 16 matrices over GF(2).
- a and
b as
16 × 16 matrices over GF(8).
- a and
b as
16 × 16 matrices over GF(8).
- a and
b as
16 × 16 matrices over GF(8).
- a and
b as
48 × 48 matrices over GF(2) - reducible over GF(8).
- All faithful irreducibles in characteristic 3.
- a and
b as
9 × 9 matrices over GF(9).
- a and
b as
9 × 9 matrices over GF(9).
- a and
b as
16 × 16 matrices over GF(3).
- a and
b as
18 × 18 matrices over GF(3).
- a and
b as
18 × 18 matrices over GF(9).
- a and
b as
18 × 18 matrices over GF(9).
- All faithful irreducibles in characteristic 17.
- a and
b as
3 × 3 matrices over GF(17) - the natural representation as O3(17).
- a and
b as
5 × 5 matrices over GF(17).
- a and
b as
7 × 7 matrices over GF(17).
- a and
b as
9 × 9 matrices over GF(17).
- a and
b as
11 × 11 matrices over GF(17).
- a and
b as
13 × 13 matrices over GF(17).
- a and
b as
15 × 15 matrices over GF(17).
- a and
b as
17 × 17 matrices over GF(17).
The representations of 2.L_{2}(17) = SL_{2}(17) available are:
- A and
B as
8 × 8 matrices over GF(9).
- A and
B as
8 × 8 matrices over GF(9).
- All faithful irreducibles in characteristic 17.
- A and
B as
2 × 2 matrices over GF(17) - the natural representation.
- A and
B as
4 × 4 matrices over GF(17).
- A and
B as
6 × 6 matrices over GF(17).
- A and
B as
8 × 8 matrices over GF(17).
- A and
B as
10 × 10 matrices over GF(17).
- A and
B as
12 × 12 matrices over GF(17).
- A and
B as
14 × 14 matrices over GF(17).
- A and
B as
16 × 16 matrices over GF(17).
The representations of L_{2}(17):2 = PGL_{2}(17) available are:
- c and
d as
permutations on 18 points.
- c and
d as
3 × 3 matrices over GF(17).
Maximal subgroups
The maximal subgroups of L_{2}(17) are as follows.
- F_{136} = 17:8.
- S_{4}.
- S_{4}.
- D_{18}.
- D_{16}.
The maximal subgroups of L_{2}(17):2 = PGL_{2}(17) are as follows.
- L_{2}(17).
- F_{272} = 17:16.
- D_{36}.
- D_{32}.
Conjugacy classes
The 11 conjugacy classes of L_{2}(17) are as follows.
- 1A: identity.
- 2A: a.
- 3A: b.
- 4A: [a, bab].
- 8A: (ab)^{3}(ab^{-1})^{2}.
- 8B: (ab)^{3}ab^{-1}.
- 9A: [a, b].
- 9B: [a, b]^{2}.
- 9C: ababab^{-1} or [a, b]^{4}.
- 17A: ab.
- 17B: (ab)^{3}.
The 19 conjugacy classes of L_{2}(17):2 = PGL_{2}(17) are as follows.
- 1A: identity.
- 2A: [c, d]^{2}.
- 3A: d.
- 4A: [c, d].
- 8A: (cd)^{6}.
- 8B: (cd)^{2}.
- 9A: (cd)^{5}cd^{-1}.
- 9B: [c, dcdcd].
- 9C: [c, dcd].
- 17AB: (cd)^{6}(cd^{-1})^{2}.
- 2B: c.
- 6A: (cd)^{4}cd^{-1}.
- 16A: (cd)^{5}.
- 16B: cd.
- 16C: (cd)^{3}.
- 16D: (cd)^{7}.
- 18A: (cd)^{7}(cd^{-1})^{2}.
- 18B: (cd)^{3}(cd^{-1})^{2}.
- 18C: cdcdcd^{-1}.
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Last updated 11th April 2000,
R.A.Wilson, R.A.Parker and J.N.Bray