ATLAS: Linear group L_{5}(2)
Order = 9999360 = 2^{10}.3^{2}.5.7.31.
Mult = 1.
Out = 2.
The page for the Dempwolff group 2^{5}.L_{5}(2) is available
here.
Standard generators
Standard generators of L_{5}(2) are a and b where
a is in class 2A, b has order 5 and ab has order 21.
Standard generators of L_{5}(2):2 are c and d where
c is in class 2C, d is in class 8B, cd has order 21
and cdcdd has order 8.
(NB: it is not possible to have d in 8C with these other properties.)
Automorphisms
An outer automorphism of L_{5}(2) of order 2 may be obtained by mapping (a, b) to (a, b^{1}).
Standard generators of L_{5}(2):2 may be obtained as follows:
c is the given automorphism and d = cab.
To return to L_{5}(2), the pair (a', b') =
(dddd, (cdd)^1cdcdcddcdcdcddcdcdd) is
equivalent to (a, b) (i.e. they are conjugate in L_{5}(2):2).
Presentations
Presentations of L_{5}(2) and L_{5}(2):2 on their standard generators are given below.
< a, b  a^{2} = b^{5} = (ab)^{21} = [a, b]^{4} = [a, b^{2}]^{2} = (ababab^{2})^{4} = 1 >.
< c, d  c^{2} = d^{8} = (cd)^{21} = (cd^{4})^{4} = [c, d]^{5} = [c, d^{2}]^{2} = (cdcdcdcd^{2})^{2} = 1 >.
Representations
The representations of L_{5}(2) available are:

Permutations on 31 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 155 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 2:

Dimension 5 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.

Dimension 5 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 24 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 3:

Dimension 30 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 124 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 155 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 5:

Dimension 30 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 123 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 155 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 280 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 7:

Dimension 30 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 94 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 31:

Dimension 29 over GF(31):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 124 over GF(31):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 251 over GF(31):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of L_{5}(2):2 available are:

Permutations on 62 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 10 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 20 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 24 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 30 over GF(9):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 30 over GF(25):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 30 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 29 over GF(31):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 30 over Z[r2]:
c and d (Magma).
Maximal subgroups
The maximal subgroups of L_{5}(2) are as follows.
The maximal subgroups of L_{5}(2):2 are as follows.
Conjugacy classes
A set of generators for the maximal cyclic subgroups of L_{5}(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 L_{5}(2):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.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to old L5(2) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 created on 6th May 1999.
Last updated 14.12.01 by RAW.
Information checked to
Level 0 on 06.05.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.