ATLAS: Linear group L_{3}(3)
Order = 5616 = 2^{4}.3^{3}.13.
Mult = 1.
Out = 2.
Standard generators
Standard generators of L_{3}(3) are
a
and b where
a has order 2,
b is in class 3B
and ab is in class 13A/B. The last condition is equivalent to:
ab has order 13
and ababb has order 4.
Wlog define 13A to be the class containing ab, and then 8B is the class containing
abababb.
Standard generators of L_{3}(3):2 are
c
and d where
c is in class 2B,
d is in class 4B
and cd is in class 13AB.
The last condition is equivalent to:
cd has order 13
and cdcdcdd has order 12.
Representations
The representations of L_{3}(3) available are
 All primitive permutation representations.

Permutations on 13 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 13 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the image of the above under an outer automorphism.

Permutations on 144 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 234 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 2 (up to Frobenius automorphisms).

Dimension 12 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

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

Dimension 26 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 3.

Dimension 3 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the natural representation.

Dimension 3 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the dual and skewsquare of the above.

Dimension 6 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the symmetric square of the natural representation.

Dimension 6 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the dual of the above.

Dimension 7 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 15b over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 27 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the Steinberg representation.
 Faithful irreducibles in characteristic 13.

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).

Dimension 16 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 26 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 26 over GF(169):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 26 over GF(169):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 39 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Faithful irreducibles in characteristic 0.
 a and
b as 12 × 12 matrices over Z.
 a and
b as 13 × 13 monomial matrices over Z.
 a and
b as 26 × 26 matrices over Z.
 a and
b as 26 × 26 matrices over Z[i2].
 a and
b as 26 × 26 matrices over Z[i2]  the dual of the above.
 a and
b as 27 × 27 matrices over Z.
 a and
b as 39 × 39 monomial matrices over Z.
 a and
b as 52 × 52 matrices over Z  reducible over Q(i2).
 a and
b as 64 × 64 matrices over Z  reducible over Q(b13) and Q(d13).
The representations of L_{3}(3):2 available are
 Faithful permutation representations, including all primitive ones.

Permutations on 26 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 imprimitive.

Permutations on 52 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 primitive.

Permutations on 117 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 primitive.

Permutations on 144 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 primitive.

Permutations on 234 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 primitive.
 All faithful irreducibles in characteristic 2.

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

Dimension 32a over GF(4):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 32b over GF(4):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 26 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 3 (up to tensoring with linear characters).

Dimension 6 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 12 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 7 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 30 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 27 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 13 (up to tensoring with linear characters).

Dimension 11 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 13 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 16 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 26 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 52 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 39 over GF(13):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
Maximal subgroups
The maximal subgroups of L_{3}(3) are as follows.
 3^2:2S4.
 3^2:2S4.
 13:3.
 S4.
The maximal subgroups of L_{3}(3):2 are as follows.
 L3(3).
 3^1+2.D8.
 2.S4.2.
 13:6.
 S4 × 2.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
Go to old L3(3) page  ATLAS version 1.
Anonymous ftp access is also available on
for.mat.bham.ac.uk.
Version 2.0 created on 13th December 2001.
Last updated 13.12.01 by RAW.
Information checked to
Level 0 on 13.12.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.