ATLAS: Linear group L_{3}(7)
Order = 1876896.
Mult = 3.
Out = S_{3}.
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
- a and
b as
permutations on 57 points.
- a and
b as
152 x 152 matrices over GF(2).
- a and
b as
55 x 55 matrices over GF(3).
- a and
b as
57 x 57 matrices over GF(3).
- a and
b as
96 x 96 matrices over GF(3).
- a and
b as
96 x 96 matrices over GF(3).
- a and
b as
96 x 96 matrices over GF(3).
- a and
b as
342 x 342 matrices over GF(3).
- a and
b as
399 x 399 matrices over GF(3).
- a and
b as
8 x 8 matrices over GF(7).
- a and
b as
10 x 10 matrices over GF(7).
- a and
b as
10 x 10 matrices over GF(7).
- a and
b as
27 x 27 matrices over GF(7).
The representations of L3(7):2 available are
- c and
d as
152 x 152 matrices over GF(2).
- c and
d as
57 x 57 matrices over GF(3).
- c and
d as
96 x 96 matrices over GF(3).
- c and
d as
8 x 8 matrices over GF(7).
- c and
d as
20 x 20 matrices over GF(7).
The representations of SL3(7) = 3.L3(7) available are
- A and
B as
3 x 3 matrices over GF(7) - the natural representation.
The representations of 3.L3(7):2 available are
- C and
D as
6 x 6 matrices over GF(7).
Maximal subgroups
The maximal subgroups of L3(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 L3(7):2 are as follows.
- L3(7).
- 7^1+2:(3 x D8).
- 2.(2 x L2(7)).2
- L2(7):2 x2.
- S3 x S4.
- 3^2:SD16.
- 19:6.
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Last updated 02.06.00
R.A.Wilson@bham.ac.uk
richard@ukonline.co.uk