ATLAS: Linear group L_{2}(16)
Order = 4080 = 2^{4}.3.5.17.
Mult = 1.
Out = 4.
The following information is available for L_{2}(16):
Standard generators of L_{2}(16) are a and b where
a has order 2, b has order 3 and ab has order 15.
Standard generators of L_{2}(16):2 are c and d where c is in class 2B, d has order 4 and cd has order 15.
Standard generators of L_{2}(16):4 are e and f where e is in class 2A, f is in class 4B/B', ef has order 8, eff has order 6 and efeffefff has order 4.
.
The outer automorphism of order 2 of L2(16) may be achieved by applying
this program to the standard generators.
The outer automorphism of order 4 of L2(16) may be achieved by applying
this program to the standard generators.
(Modulo inner automorphisms, this is the Frobenius automorphism *2.)
Presentations for L_{2}(16), L_{2}(16):2 and L_{2}(16):4 in terms of their standard generators are given below.
< a, b  a^{2} = b^{3} = (ab)^{15} = ((ab)^{5}(ab^{1})^{3})^{2} = 1 >.
< c, d  c^{2} = d^{4} = [c, dcd]^{2} = cd(cdcd^{2})^{2}(cd^{2}cd)^{2}cdcd^{1} = 1 >.
< e, f  e^{2} = f^{4} = (ef)^{8} = (ef^{2})^{6} = (ef)^{3}(ef^{1})^{3}(ef)^{2}(ef^{1})^{2}(ef)^{2}ef^{2} = 1 >.
The representations of L_{2}(16) available are

Permutations on 17 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

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

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

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

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

Dimension 16 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles over GF(17):

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).

Dimension 34 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 reducible over GF(289).

Dimension 68 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 becomes dimension 17 over GF(17^4).
The representations of L_{2}(16):2 available are

Dimension 4 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of L_{2}(16):4 available are
 All faithful irreducibles in characteristic 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(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the Steinberg module.

Dimension 32 over GF(2):
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 60 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 60 over GF(9):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 16 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

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

Dimension 60 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 60 over GF(5):
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).

Dimension 34 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 68 over GF(17):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The maximal subgroups of L_{2}(16) are as follows.
 2^{4}:15._{ }
 A_{5}.
 D_{34}.
 D_{30}.
The maximal subgroups of L_{2}(16):2 are as follows.
 L_{2}(16).
 2^{4}:(3 × D_{10}).
 A_{5} × 2.
 F_{68} = 17:4.
 D_{10} × S_{3}.
The maximal subgroups of L_{2}(16):4 are as follows.
 L_{2}(16):2.
 2^{4}:15:4._{ }
 (A_{5} × 2).2.
 F_{136} = 17:8.
 5:4 × S_{3}.
A set of generators for the maximal cyclic subgroups of L2(16) can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
A set of generators for the maximal cyclic subgroups of L2(16):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.
A set of generators for the maximal cyclic subgroups of L2(16):4 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 classes of L2(16).4 are chosen so that efefff = [e,f] is in class 17EFGH.
Go to main ATLAS (version 2.0) page.
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Anonymous ftp access is also available.
See here for details.
Version 2.0 file created on 24th January 2002,
from Version 1 file last updated on 08.10.98.
Last updated 29.01.02 by RAW.
Information checked to
Level 0 on 24.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.