ATLAS: Linear group L_{3}(8)
Order = 16482816 = 2^{9}.3^{2}.7^{2}.73.
Mult = 1.
Out = 6.
The following information is available for L_{3}(8):
Standard generators
Standard generators of L_{3}(8) are
a and b where
a has order 2, b has order 3
and ab has order 21.
Standard generators of L_{3}(8):2 are c and
d where c is in class 2B, d
has order 3, cd has order 8, cdcdd has order
63 and cdcdcdd has order 8.
Standard generators of L_{3}(8):3 are e and
f where e has order 2, f has
order 12, ef has order 21, efeff has order 7,
efefeff has order 12, efefefeff has
order 12 and efefeffefeffeffeff has order 9.
Standard generators of L_{3}(8):6 are g and
h where g is in class 2A, h is in
class 18D, gh has order 24, ghghh has order 14,
ghghghh has order 3 and ghghghghghhghghhghh has
order 6.
Automorphisms
An automorphism of L_{3}(8) of order 2 may be obtained by mapping (a,b) to (a,b^{1}).
An automorphism of L_{3}(8) of order 3 may be obtained by applying
this program to the standard generators.
An automorphism of L_{3}(8) of order 6 may be obtained by applying
this program to the standard generators.
Black box algorithms
To find standard generators for L_{3}(8):
 Find an element of order 2, x say, by taking a suitable power of
an element of even order.
 Find an element of order 3, y say, by taking a suitable power of an
element of order divisible by 3.
 Find conjugates a of x and b of y such that
ab has order 21.
 Now a and b are standard generators of L_{3}(8).
To find standard generators for L_{3}(8).2:
 Find any element of order 6 or 18. This powers up to x in class 2B.
 Find an element of order 3, y say, by taking a suitable power of an
element of order divisible by 3.
 Find conjugates c of x and d of y such that
cd has order 8, cdcdd has order 63 and cdcdcdd has order 8.
 Now c and d are standard generators of L_{3}(8).2.
To find standard generators for L_{3}(8).3:
 Find an element of order 2, x say, by taking a suitable power of an
element of even order.
 Find an element y of order 12.
 Find conjugates e of x and f of y such that
ef has order 21, efeff has order 7, efefeff has order 12 and efefefeff
has order 12.
 If efefeffefeffeffeff does not have order 9 then replace f
with f^{1}.
 Now e and f are standard generators of L_{3}(8).3.
To find standard generators for L_{3}(8).6:
 Find an element in class 2A, x say, by taking a suitable power of an
element of order divisible by 4.
 Find an element y of order 18.
 If the product xy does not have order 24 then go back one step.
 Find conjugates g of x and h of y such that
ghghh has order 14 and ghghghh has order 3.
 If ghghghghhghhghhghhghh does not have order 6 then replace h
with h^{1}.
 Now g and h are standard generators of L_{3}(8).6.
Representations
The representations of L_{3}(8) available are:
 Permutations on 73 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 Permutations on 73 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 Permutations on 56064 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 Permutations on 75264 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 Permutations on 98112 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 Some irreducibles in characteristic 2:

Dimension 3 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 8 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP)
,
a and b (Magma).

Dimension 9 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 9 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 24 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 24 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 27 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 27 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 64 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 72 over GF(8):
a and
b (Meataxe)
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 72 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 192 over GF(8):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 512 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 Some irreducibles in characteristic 3:

Dimension 72 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 511 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 657 over GF(27):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 Some irreducibles in characteristic 7:

Dimension 72 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 511 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary).
a and
b (GAP),
a and b (Magma).

Dimension 511 over GF(343):
a and
b (Meataxe),
a and b (Magma).

Dimension 512 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
 Some irreducibles in characteristic 73:

Dimension 71 over GF(73):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 441 over GF(73):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 511 over GF(73):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 511 over GF(73);
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
The representations of L_{3}(8):2 available are:
 Permutations on 657 points:
c and
d (Meataxe),
c and
d (Meataxe),
c and
d (GAP),
c and d (Magma).
 Permutations on 4672 points:
c and
d (Meataxe),
c and
d (Meataxe),
c and
d (GAP),
c and d (Magma).
 Permutations on 56064 points:
c and
d (Meataxe),
c and
d (Meataxe),
c and
d (GAP),
c and d (Magma).
 Permutations on 75264 points:
c and
d (Meataxe),
c and
d (Meataxe),
c and
d (GAP),
c and d (Magma).
 Permutations of 98112 points:
c and
d (Meataxe),
c and
d (Meataxe),
c and
d (GAP),
c and d (Magma).

Some irreducibles in characteristic 2:
 Dimension 6 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 8 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 18 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 18 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 48 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 48 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 54 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 54 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 64 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
 Dimension 144 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 144 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 384 over GF(8):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 512 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Some irreducibles in characteristic 3:

Dimension 72 over GF(9):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 511 over GF(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Some irreducibles in characteristic 7:

Dimension 72 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 511 over GF(7):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 512 over GF(7);
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Some irreducibles in characteristic 73:

Dimension 71 over GF(73):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 441 over GF(73):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 511 over GF(73):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).

Dimension 511 over GF(73):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP),
c and d (Magma).
The representationsof L_{3}(8):3 available are:
 Permutations on 73 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
 Permutations on 73 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
 Permutations on 56064 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
 Permutations on 75264 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
 Permutations on 98112 points — imprimitive:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Some irreducibles in charactersitic 2:

Dimension 9 over GF(2);
d and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 24 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 27 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 27 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 72 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 72 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 27 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 81 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 192 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 216 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 216 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 576 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 512 over GF(2):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Some irreducibles in characteristic 3:

Dimension 72 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 511 over GF(3):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
Some irreducibles in characteristic 7:

Dimension 72 over GF(7):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 511 over GF(7):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 512 over GF(7):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Some irreducibles in charactersitic 73:

Dimension 71 over GF(73):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 441 over GF(73):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).

Dimension 511 over GF(73):
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
The representations of L_{3}(8):6 available are:
 Permutations on 657 points:
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).
 Permutations on 4672 points:
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).
 Permutations on 56064 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
 Permutations on 75264 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
 Permutations on 98112 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP),
e and f (Magma).
 Some irreducibles in characteristic 2:

Dimension 18 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 24 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 54 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 54 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 144 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 144 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Mamga).

Dimension 54 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 162 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 192 over GF(2):
g and
h (Meataxe)
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 432 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 432 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 1152 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 512 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Some irreducibles in characteristic 3:
 Dimension 72 over GF(9):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 511 over GF(3):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Some irreducibles in characteristic 7:
 Dimension 72 over GF(7):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 511 over GF(7):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 512 over GF(7):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Some irreducibles in characteristic 73:
 Dimension 71 over GF(73):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 441 over GF(73):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).

Dimension 511 over GF(73):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP),
g and h (Magma).
Maximal Subgroups
The maximal subgroups of L_{3}(8) are as follows.
 2^6:(7xL2(8)), with generators a^(ab), (ab)^((abb)^8).
 2^6:(7xL2(8)), with generators a^((ab)^6), (ab)^(abb).
 7^2:S3, with generators a^((ab)^4), b^(abb).
 73:3, with generators b^((ab)^5), b^((abb)^14).
 L2(7), with generators a^((ab)^5), b^((abb)^3).
The maximal subgroups of L_{3}(8):2 are as follows.
 L3(8), with generators ((cd)^4)^(cd), d^(cdd).
 2^(3+6):7^2:2, with generators c^(cd),(cdcdcdcddcdcdcddcddcdcdd)^((cdd)^4).
 D14xL2(8), with generators c^(cd), ((cdcdcddcddcdcdcddcdcddcdd)^6)^((cdd)^2).
 7^2:D12, with generators ((cd)^4)^(cdd)^4, ((cdcdcdcddcdd)^3)^((cdcdd)^3).
 73:6, with generators c^((cdcdd)^4), b^(cdd).
 L2(7):2, with generators ((cd)^2)^(cd), ((cdcdcdcddcdd)^3)^((cdd)^6).
The maximal subgroups of L_{3}(8):3 are as follows.
 L3(8), with generators (f^6)^(ef), (f^3)^((eff)^2).
 2^6:(7xL2(8)):3, with generators (efefefeffeffefefefeffeffefefeff)^((ef)^7), (f^2)^((eff)^3).
 2^6:(7xL2(8)):3, with generators (efefefeffeffefefefeffeffefefeff)^((ef)^2), (f^2)^((eff)^2).
 7^2:(3xS3), with generators (f^6)^((ef)^12), (f^2)^((eff)^3).
 73:9, with generators ((efefefefeffeffeffeff)^483)^((ef)^2), (efefefefeffeff)^((efefeff)^6).
 L2(7)x3, with generators (f^6)^(ef), ((efefefefeffeff)^3)^(eff) — almost certainly wrong.
The maximal subgroups of L_{3}(8):6 are as follows.
 L3(8).3, with generators ((gh)^12)^(gh), ((gh)^4)^((ghh)^5).
 L3(8).2, with generators ((gh)^12)^(gh), (ghghghhghghhghh)^(ghh).
 2^(3+6):7^2:6, with generators ((ghghghghhghhghhghghh)^2)^((gh)^2), (ghghghghghhghghhghh)^((ghh)^7).
 (D14xL2(8)):3, with generators ((ghghghghhghhghhghghh)^2)^((gh)^2), (ghghghghghhghghhghh)^((ghh)^6).
 7^2:(6xS3), with generators ((ghghghghhghhghhghghh)^7)^((gh)^5), (ghghghghghhghghhghh)^((ghh)^10).
 73:18, with generators ((ghghghghhghhghh)^9)^((gh)^6), h^((ghh)^4).
 L2(7):2x3, with generators ((ghghhghghghhghhghhghghh)^3)^((gh)^16), (ghghghghghhghghhghh)^((ghh)^23).
Conjugacy Classes
A set of generators for the maximal cyclic subgroups of L_{3}(8)
can be obtained
by running this program on the standard
generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
A set of generators for the maximal cyclic subgroups of L_{3}(8):2 can be
obtained by running this program on the
standard generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
A set of generators for the maximal cyclic subgroups of L_{3}(8):3 can be
obtained by running this program on the
standard generators. All conjugacy classes can therefore be obtained as suitable
powers of these elements.
A set of generators for the maximal cyclic subgroups of L_{3}(8):6 can
be obtained by running this program on the standard generators.
All conjugacy classes can therefore be found as suitable powers of these elements.
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Version 2.0 created on 10th February 2004.
Last updated 17.08.04 by RAA; slight modification on 06.07.11 by JNB.
Information checked to
Level 0 on 26.02.04 by RAA.
R.A.Wilson, R.A.Parker and J.N.Bray.