ATLAS: Linear group L3(8)
Order = 16482816 = 29.32.72.73.
Mult = 1.
Out = 6.
The following information is available for L3(8):
Standard generators
Standard generators of L3(8) are
a and b where
a has order 2, b has order 3
and ab has order 21.
Standard generators of L3(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 L3(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 L3(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 L3(8) of order 2 may be obtained by mapping (a,b) to (a,b-1).
An automorphism of L3(8) of order 3 may be obtained by applying
this program to the standard generators.
An automorphism of L3(8) of order 6 may be obtained by applying
this program to the standard generators.
Black box algorithms
To find standard generators for L3(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 L3(8).
To find standard generators for L3(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 L3(8).2.
To find standard generators for L3(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 L3(8).3.
To find standard generators for L3(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 L3(8).6.
Representations
The representations of L3(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 L3(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 L3(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 L3(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 L3(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 L3(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 L3(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 L3(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 L3(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 L3(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 L3(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 L3(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.
Go to main ATLAS (version 2.0) page.
Go to linear groups page.
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.