ATLAS: Linear group L_{3}(7)
Order = 1876896.
Mult = 3.
Out = S_{3}.
The following information is available for L_{3}(7):
Standard generators
Standard generators of L3(7) are
a
and b where
a has order 2,
b has order 3,
ab has order 19,
and ababb has order 6.
Standard generators of 3.L3(7) are preimages
A
and B where
A has order 2 and
AB has order 19.
Standard generators of L3(7).2 are
c
and d where
c is in class 2B,
d is in class 4B,
cd has order 19,
and cdcdd has order 8.
Standard generators of 3.L3(7).2 are preimages
C
and D where
CD has order 19.
Representations
The representations of L3(7) available are

Permutations on 57 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 152 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 3:

Dimension 55 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 57 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 96 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 96 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 96 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 342 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 399 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some irreducibles in characteristic 7:

Dimension 8 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of L3(7):2 available are
 Some irreducibles in characteristic 2:

Dimension 56 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 152 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 342 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some irreducibles in characteristic 3:

Dimension 55 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 57 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 96 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 192 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 342 over GF(9):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 342 over GF(9):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 399 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 Some irreducibles in characteristic 7:

Dimension 8 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 20 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 27 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 37 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 56 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 70 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 308 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 343 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of SL3(7) = 3.L3(7) available are

Dimension 3 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 the natural representation.
The representations of 3.L3(7):2 available are

Dimension 6 over GF(7):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
Maximal subgroups
The maximal subgroups of L_{3}(7) are as follows.
 7^2:2.L2(7):2.
 7^2:2.L2(7):2.
 L2(7):2.
 L2(7):2.
 L2(7):2.
 (3 x A4)2.
 3^2:Q8.
 19:3
The maximal subgroups of L_{3}(7).2 are as follows.
 L3(7), with generators
here.
 7^1+2:(3 x D8), with generators
here.
 2.(2 x L2(7)).2
 L2(7):2 x2.
 S3 x S4.
 3^2:SD16.
 19:6.
Conjugacy classes
A set of generators for the maximal cyclic subgroups of L_{3}(7)
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_{3}(7):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 L3(7) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 created on 22nd March 2001.
Last updated 14.12.01 by RAW.
Information checked to
Level 0 on 14.12.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.