ATLAS: Linear group L3(3)
Order = 5616 = 24.33.13.
Mult = 1.
Out = 2.
Standard generators
Standard generators of L3(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 L3(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 L3(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 skew-square 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 L3(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 L3(3) are as follows.
- 3^2:2S4.
- 3^2:2S4.
- 13:3.
- S4.
The maximal subgroups of L3(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.