ATLAS: Linear group L_{2}(8)
Order = 504 = 2^{3}.3^{2}.7.
Mult = 1.
Out = 3.
The following information is available for L_{2}(8):
Standard generators
Standard generators of L_{2}(8) are a and b where
a has order 2, b has order 3 and ab has order 7.
Standard generators of L_{2}(8):3 are c and d where
c has order 2, d has order 3, cd has order 9 and cdcdcddcdcddcdd has order 7. The last condition is equivalent to: cdcdcddcddcdcdd has order 2. Note that these conditions imply that
d is conjugate to the field automorphism that squares the matrix
entries in the natural representation of L_{2}(8).
Black box algorithms
To find standard generators for L_{2}(8):
- Find any elements x of order 2 and y of order 3.
- Find a conjugate a of x and a conjugate b of y whose product has order 7.
- The elements a and b are now standard generators of L_{2}(8).
To find standard generators for L_{2}(8).3:
- Find any element of order 6. This cubes to x of order 2 and squares
to y of order 3.
- Find a conjugate c of x and a conjugate d of
y whose product has order 9.
- If cdcdcddcdcddcdd has order 2 then invert d.
- The elements c and d are now standard generators of
L_{2}(8):3.
Automorphisms
An outer automorphism of L_{2}(8) of order 3 may be obtained by mapping (a, b) to (a, b^{ababba}).
To obtain our standard generators for L_{2}(8):3 we may take c = babb and d to be the above automorphism.
Conversely, we may take a = cd^{-1}cdcd^{-1}cd^{-1}cdcdcd^{-1}cdc and b = cdcdcd^{-1}cd^{-1}cd^{-1}cdcd^{-1}cd. Note also that a' = c and b' = d^{-1}(cd)^{3}d are equivalent under an automorphism to (a, b).
Presentations
Presentations for L_{2}(8) and L_{2}(8):3 = R(3) in terms of their standard generators are given below.
< a, b | a^{2} = b^{3} = (ab)^{7} = (ababab^{-1}ababab^{-1}ab^{-1})^{2} = 1 >.
< c, d | c^{2} = d^{3} = (cd)^{9} = [c, d]^{9} = (cdcdcd^{-1}cd^{-1}cdcd^{-1})^{2} = 1 >.
Representations
The representations of L_{2}(8) available are
- All primitive permutation representations.
- a and
b as
permutations on 9 points.
- a and
b as
permutations on 28 points.
- a and
b as
permutations on 36 points.
- a and
b as
2 × 2 matrices over GF(8) - the natural representation.
The representations of L_{2}(8):3 available are
- All faithful primitive permutation representations.
- c and
d as
permutations on 9 points.
- c and
d as
permutations on 28 points.
- c and
d as
permutations on 36 points.
- All faithful irreducible representations in characteristic 2 whose character appears in the ABC.
- c and
d as
6 × 6 matrices over GF(2).
- c and
d as
8 × 8 matrices over GF(2).
- c and
d as
12 × 12 matrices over GF(2).
- Both faithful irreducible representations in characteristic 3.
- c and
d as
7 × 7 matrices over GF(3) - the natural representation as R(3).
- c and
d as
27 × 27 matrices over GF(3).
- All faithful irreducible representations in characteristic 7 whose character appears in the ABC.
- c and
d as
7 × 7 matrices over GF(7).
- c and
d as
8 × 8 matrices over GF(7).
- c and
d as
21 × 21 matrices over GF(7).
- All faithful irreducible representations in characteristic 0 whose character appears in the ATLAS.
Maximal subgroups
The maximal subgroups of L_{2}(8) are as follows.
The maximal subgroups of L_{2}(8):3 are as follows.
- L2(8).
- 2^3:7:3.
- 9:6.
- 7:6.
Conjugacy classes
The following are conjugacy class representatives of L_{2}(8).
- 1A: identity.
- 2A: a.
- 3A: b.
- 7A: ab.
- 7B: abab.
- 7C: ababab or abababab.
- 9A: ababb.
- 9B: ababbababb.
- 9C: abababbabb or (ababb)^4.
The following are conjugacy class representatives of L_{2}(8):3.
- 1A: identity.
- 2A: c.
- 3A: cdcdcd.
- 7ABC: cdcdcddcdd.
- 9ABC: cdcdd.
- 3B: d.
- 6A: cdcdcdd.
- 9D: cd.
- 3B': dd.
- 6A': cdcddcdd.
- 9D': cdd.
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Last updated 29th October 1999,
R.A.Wilson, R.A.Parker and J.N.Bray