ATLAS: Linear group L_{2}(17)
Order = 2448 = 2^{4}.3^{2}.17.
Mult = 2.
Out = 2.
The following information is available for L_{2}(17):
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 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 >.
The representations of L_{2}(17) available are:

Permutations on 18 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 2 and over GF(2).

Dimension 8 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 8 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 16 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 16 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 16 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 16 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 48 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(8).
 All faithful irreducibles in characteristic 3.

Dimension 9 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 16 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 18 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 17.

Dimension 3 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation as O3(17).

Dimension 5 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 7 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 11 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 13 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 17 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.L_{2}(17) = SL_{2}(17) available are:

Dimension 8 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 8 over GF(9):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 17.

Dimension 2 over GF(17):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 the natural representation.

Dimension 4 over GF(17):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 6 over GF(17):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 8 over GF(17):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10 over GF(17):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 12 over GF(17):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 14 over GF(17):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 16 over GF(17):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of L_{2}(17):2 = PGL_{2}(17) available are:

Permutations on 18 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 3 over GF(17):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2L_{2}(17).2 available are:

Dimension 16 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
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}.
A set of generators for the maximal cyclic subgroups of L2(17) 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 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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Anonymous ftp access is also available.
See here for details.
Version 2.0 file created on 11th April 2000.
Last updated 21.01.02 by RAW.
Information checked to
Level 0 on 18.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.