ATLAS: Linear group L_{2}(3)
Order = 1092.
Mult = 2.
Out = 2.
The following information is available for L_{2}(13):
Standard generators of L2(13) are a
and b where
a has order 2, b has order 3
and ab has order 13.
Standard generators of the double cover 2.L2(13) = SL2(13) are preimages
A
and B where
B has order 3
and AB has order 13.
Standard generators of L2(13).2 are c
and d where
c has order 2 (necessarily class 2B), d has order 4
and cd has order 13.
Standard generators of the double cover 2.L2(13).2 are preimages
C
and D where
CD has order 13.
The outer automorphism of L2(13) may be achieved by applying
this program to the standard generators.
The representations available are

Permutations on 14 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All irreducibles in characteristic 2:

Dimension 6 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 6 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 12 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 12 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 12 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 13 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 12 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All irreducibles in characteristic 13:

Dimension 3 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 5 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 7 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 9 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 11 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 13 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
The representations of 2.L2(13) available are
 All irreducibles in characteristic 3:

Dimension 6 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 6 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 12 over GF(27):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 12 over GF(27):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 12 over GF(27):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 14 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All irreducibles in characteristic 7:

Dimension 6 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 6 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 14 over GF(7):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 14 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 14 over GF(49):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All irreducibles in characteristic 13:

Dimension 2 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 the natural representation.

Dimension 4 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 6 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 8 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 10 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 12 over GF(13):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of L2(13).2 available are

Permutations on 14 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 78 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 91 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 12 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 3 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of 2L2(13).2 available are

Dimension 12 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of L2(13) are as follows.
The maximal subgroups of L2(13):2 are as follows.
A set of generators for the maximal cyclic subgroups of L2(13) can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
In the double cover, the element ABABABB is in class +6A, for compatibility
with the ABC.
A set of generators for the maximal cyclic subgroups of L2(13):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 L2(13) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 file created on 21st January 2002,
from Version 1 file last updated on 13.03.96.
Last updated 29.01.02 by RAW.
Information checked to
Level 0 on 21.01.02 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.