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.
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.
- a and
b as
permutations on 13 points.
- a and
b as
permutations on 13 points - the image of the above under an outer automorphism.
- a and
b as
permutations on 144 points.
- a and
b as
permutations on 234 points.
- Faithful irreducibles in characteristic 2.
- a and
b as
12 × 12 matrices over GF(2).
- a and
b as
16 × 16 matrices over GF(16).
- a and
b as
26 × 26 matrices over GF(2).
- All faithful irreducibles in characteristic 3.
- a and
b as
3 × 3 matrices over GF(3) - the natural representation.
- a and
b as
3 × 3 matrices over GF(3) - the dual and skew-square of the above.
- a and
b as
6 × 6 matrices over GF(3) - the symmetric square of the natural representation.
- a and
b as
6 × 6 matrices over GF(3) - the dual of the above.
- a and
b as
7 × 7 matrices over GF(3).
- a and
b as
15 × 15 matrices over GF(3).
- a and
b as
15 × 15 matrices over GF(3) - the dual of the above.
- a and
b as
27 × 27 matrices over GF(3) - the Steinberg representation.
- Faithful irreducibles in characteristic 13.
- a and
b as
11 × 11 matrices over GF(13).
- a and
b as
13 × 13 monomial matrices over GF(13).
- a and
b as
16 × 16 matrices over GF(13).
- 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.
- c and
d as
permutations on 26 points - imprimitive.
- c and
d as
permutations on 52 points - primitive.
- c and
d as
permutations on 117 points - primitive.
- c and
d as
permutations on 144 points - primitive.
- c and
d as
permutations on 234 points - primitive.
- c and
d as
12 × 12 matrices over GF(2).
- c and
d as
6 × 6 matrices over GF(3).
- c and
d as
11 × 11 matrices over GF(13).
- c and
d as
13 × 13 matrices over GF(13).
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.
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Last updated 09.07.98
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk