ATLAS: Linear group L_{7}(2)
Order = 163849992929280 =
2^{21}.3^{4}.5.7^{2}.31.127.
Mult = 1.
Out = 2.
The following information is available for L_{7}(2):
Standard generators of L_{7}(2) are a and b where
a is in class 2A, b has order 7 (necessarily class 7E),
ab has order 127 and ababababb has order 7.
Type I standard generators of L_{7}(2):2 are c and d
where c is in class 2A, d is in class 12H (centr.order 144),
cd has order 14 and cdcdd has order 12.
Type II standard generators of L_{7}(2):2 are e and f
where e is in class 2D, f is in class 8E/F (centr.order 92160),
ef has order ?? and efeff has order ??.
NB: Class 2A is the class of transvections (in a natural 7dimensional
representation) and class 7E is the class of 7cycles, and is the unique class
of elements of order 7 with the smallest centraliser (size 49). The centraliser
orders quoted for L_{7}(2):2 are for the
L_{7}(2):2centralisers, not the L_{7}(2)centralisers (unlike
the ATLAS). Class 2D is the unique class of outer involutions.
An outer automorphism of L_{7}(2) of order 2 may be obtained by
mapping (a, b) to
(a, b^{1}).
Let u be the given automorphism. Then we may obtain standard generators
of L_{7}(2):2 as follows:
Type I generators are given by
c = a and d =
(ab)^{4}u(ab)^{10}u(ab)^{3}u(ab)^{1}.
Type II generators are given by
e = u and f = uab.
We can return to L_{7}(2) by letting
a = fe[e, f]^{4} and
b = [e, f]^{4}. (This one is the exact inverse
of the above, not merely an inverse up to automorphisms.)
Presentations of L_{7}(2) and L_{7}(2):2 on their standard generators are given below.
< a, b  a^{2} = b^{7} =
[a, b]^{4} =
[a, b^{2}]^{2} =
[a, b^{3}]^{2} =
((ab)^{5}b^{4})^{15} =
(ababab^{2})^{4} = 1 >.
< c, d  c^{2} = d^{12} =
(cd)^{14} = (cd^{6})^{4} =
[c, d^{2}]^{3} =
(cd^{2}cd^{2}cd^{6})^{3} =
[c, d^{2}cdcdcd^{2}]^{2} =
[c, dcd^{2}]^{2} =
[c, dcd^{3}]^{2} =
[c, dcdcd^{2}cdcd] = 1 >.
< e, f  e^{2} = f^{8} =
(ef^{4})^{4} = [e, f]^{7} =
[e, f^{2}]^{2} =
(efef^{2}ef^{4})^{15} =
(efef^{2}efef^{2})^{2} =
((ef)^{6}f^{3})^{2} = 1, more relations? >.
Just for comparison (for now), L_{5}(2):2 on its standard generators.
< c, d  c^{2} = d^{8} = (cd)^{21} = (cd^{4})^{4} = [c, d]^{5} = [c, d^{2}]^{2} = (cdcdcdcd^{2})^{2} = 1 >.
The representations of L_{7}(2) available are:

Permutations on 127a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 127b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 2 (all up to dimension 1000).

Dimension 7a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.

Dimension 7b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 its dual.

Dimension 21a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 21b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 35b over GF(2):
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).
 the adjoint representation.

Dimension 112a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 112b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 133b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 175a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 175b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 224a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 224b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 392a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 448a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 448b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 469a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 469b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 707a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 707b over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 736a over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 A faithful irreducible in characteristic 0
 Dimension 126 over Z:
a and b (GAP).
The representations of L_{7}(2):2 available are:

Permutations on 254 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
 imprimitive.
 All irreducibles over GF(2) in dimension less than 1000:

Dimension 14 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 42 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 48 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 70 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 224 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 266 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 350 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 392 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 448 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 736 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 896 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).

Dimension 938 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
The maximal subgroups of L_{7}(2) are as follows (roughly).
The maximal subgroups of L_{7}(2):2 are as follows (roughly).
We have not yet calculated the class representatives for L_{7}(2)
[117 classes] or L_{7}(2):2 [114 classes].
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
There is no old L7(2) page in the ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 6th June 2002.
Last updated 31.03.08 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.