ATLAS: Linear group L_{5}(2)
Order = 9999360 = 2^{10}.3^{2}.5.7.31.
Mult = 1.
Out = 2.
Dempwolff group 2^{5}.L_{5}(2) (non-split extension)
Order = 319979520 = 2^{15}.3^{2}.5.7.31.
Mult = 1.
Out = 1.
Standard generators
Standard generators of L_{5}(2) are a and b where
a is in class 2A, b has order 5 and ab has order 21.
Standard generators of the Dempwolff group 2^{5}.L_{5}(2) are preimages A and B such that A has order 2, B has order 5, ABABB has order 10, ABABABBBB has order 28 and ABABABBBABB has order 5.
(NB: All involutions of 2^{5}.L_{5}(2) not in the normal 2^{5} map onto class 2A of L_{5}(2).)
Standard generators of L_{5}(2):2 are c and d where
c is in class 2C, d is in class 8B, cd has order 21
and cdcdd has order 8.
(NB: it is not possible to have d in 8C, with these other properties.)
Automorphisms
An outer automorphism of L_{5}(2) of order 2 may be obtained by mapping (a, b) to (a, b^{-1}).
Standard generators of L_{5}(2):2 may be obtained as follows:
c is the given automorphism and d = cab.
To return to L_{5}(2), the pair (a', b') =
(dddd, (cdd)^-1cdcdcddcdcdcddcdcdd) is
equivalent to (a, b) (i.e. they are conjugate in L_{5}(2):2).
Presentations
Presentations of L_{5}(2), 2^{5}.L_{5}(2) and L_{5}(2):2 on their standard generators are given below.
< a, b | a^{2} = b^{5} = (ab)^{21} = [a, b]^{4} = [a, b^{2}]^{2} = (ababab^{-2})^{4} = 1 >.
< A, B | A^{2} = B^{5} = (AB)^{21} = [A, B^{2}]^{4} = [A, B^{-2}AB^{-2}][A, B]^{3} = (ABABAB^{-2})^{2}(ABAB^{-1}AB)^{2}(AB^{-1})^{2} = 1 >.
< c, d | c^{2} = d^{8} = (cd)^{21} = (cd^{4})^{4} = [c, d]^{5} = [c, d^{2}]^{2} = (cdcdcdcd^{-2})^{2} = 1 >.
Representations
The representations of L_{5}(2) available are:
- a and
b as
permutations on 31 points.
- a and
b as
permutations on 155 points.
- a and
b as
5 × 5 matrices over GF(2) - the natural representation.
- a and
b as
30 × 30 matrices over GF(3).
- a and
b as
124 × 124 matrices over GF(3).
- a and
b as
30 × 30 matrices over GF(5).
- a and
b as
123 × 123 matrices over GF(5).
- a and
b as
30 × 30 matrices over GF(7).
- a and
b as
94 × 94 matrices over GF(7).
- a and
b as
29 × 29 matrices over GF(31).
- a and
b as
124 × 124 matrices over GF(31).
The representations of 2^{5}.L_{5}(2) available are:
- A and
B as
permutations on 7440 points - on the cosets of C(A) = 2^{4}.2^{4}.L_{3}(2) - CFs of rep are 1+30+30+124+217+280+868+930+1240+3720.
- A and
B as
permutations on 7440 points.
- A and
B as
permutations on 7440 points.
- A and
B as
248 × 248 matrices over GF(3).
- A and
B as
248 × 248 matrices over GF(5).
- A and
B as
248 × 248 matrices over GF(7).
- A and B as
248 × 248 matrices over Z.
- This last representation is also available in MeatAxe format as
A and
B .
The representations of L_{5}(2):2 available are:
- c and
d as
permutations on 62 points.
- c and
d as
10 × 10 matrices over GF(2).
- c and
d as
30 × 30 matrices over GF(7).
- c and
d as
29 × 29 matrices over GF(31). This representation can be used to
distinguish the classes 8B and 8C, which have traces 14 and 15 respectively.
- c and d as
30 × 30 matrices over Z[r2].
Maximal subgroups
The maximal subgroups of L_{5}(2) are as follows.
The maximal subgroups of L_{5}(2):2 are as follows.
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Last updated 25th March 1999,
R.A.Wilson, R.A.Parker and J.N.Bray