ATLAS: Linear group L_{2}(49)
Order = 58800 = 2^{4}.3.5^{2}.7^{2}.
Mult = 2.
Out = 2^{2}.
Standard generators
Standard generators of L_{2}(49) are a
and b where
a has order 2, b has order 3,
ab has order 25 and ababb has order 24.
Standard generators of the double cover 2.L_{2}(49) = SL_{2}(49)
are preimages A and B where
A has order 3 and AB has order 25.
Automorphisms
An outer automorphism of type 2a of order 2 may be obtained by mapping (a, b) to (a, bb).
Presentations
Presentations for L_{2}(49) ... in terms of their standard generators are given below.
< a, b | a^{2} = b^{3} = (ababab^{-1})^{4} = (ab)^{6}(ab^{-1})^{2}(ab)^{3}(abab^{-1})^{2}(ab)^{4}(ab^{-1})^{2} = 1 >.
Representations
The representations of L_{2}(49) available are:
- All primitive permutation representations.
- a and
b as
permutations on 50 points.
- a and
b as
permutations on 175 points.
- a and
b as
permutations on 175 points.
- a and
b as
permutations on 980 points.
- a and
b as
permutations on 980 points.
- a and
b as
permutations on 1176 points.
- a and
b as
permutations on 1225 points.
- a and
b as
3 × 3 matrices over GF(49) - the natural representation as O3(49).
- a and
b as
3 × 3 matrices over GF(49).
- a and
b as
4 × 4 matrices over GF(7).
The representations of 2.L_{2}(49) = SL_{2}(49) available are:
- A and
B as
2 × 2 matrices over GF(49) - the natural representation as SL2(49).
- A and
B as
2 × 2 matrices over GF(49).
The representations of L_{2}(49):2_{1} = PGL_{2}(49) available are:
Maximal subgroups
The maximal subgroups of L_{2}(49) include the following:
- 7^{2}:24.
Order: 1176.
Index: 50.
- PGL_{2}(7).
Order: 336.
Index: 175.
- PGL_{2}(7).
Order: 336.
Index: 175.
- A_{5}.
Order: 60.
Index: 980.
- A_{5}.
Order: 60.
Index: 980.
- D_{50}.
Order: 50.
Index: 1176.
- D_{48}.
Order: 48.
Index: 1225.
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Last updated 8th October 1998,
R.A.Wilson, R.A.Parker and J.N.Bray