ATLAS: Unitary group U3(8)
Order = 5515776 = 29.34.7.19.
Mult = 3.
Out = 3 × S3.
The following information is available for U3(8):
Standard generators of U3(8) are
a and b where a has order 2,
b has order 3 (necessarily in class 3C)
and ab has order 19.
Standard generators of 3.U3(8) = SU3(8) are
preimages A and B where A has order 2
and AB has order 19.
Standard generators of U3(8):2 are
c and d where c is in class 2B,
d is in class 3C,
cd has order 8
and cdcdcddcdcddcdd has order 9.
Standard generators of 3.U3(8):2 are
preimages C and D where CDCDD has order 19.
Standard generators of U3(8):31 are
e and f where e has order 2,
f is in class 3D/E/F/D'/E'/F',
ef has order 12, efeff has order 7
and efefeffefeffeff has order 7.
These conditions distinguish classes 3D/E/F and 3D'/E'/F'.
Standard generators of 3.U3(8):31 are
preimages E and F where E has order 2 and
F has order 3.
Standard generators of U3(8):6 are
g and h where g is in class 2B,
h is in class 3D/D'/EF/EF' (i.e. an outer element of order 3),
gh has order 18,
ghghh has order 19
and ghghghhghghhghh has order 9.
These conditions force h to lie in a particular class of elements
of order 3, and we label this class as 3D.
Standard generators of 3.U3(8):6 are
preimages G and H where ...??... .
Standard generators for U3(8) may be obtained from those of U3(8):6 as
(ghghghhgh)6,
(gh)6.
Standard generators of U3(8):32 are
i and j where i has order 2,
j is in class 3G/G' and ij has order 9.
Standard generators of [any] 3.U3(8).32 are
preimages I and J where I has order 2 and ...??... .
Standard generators of U3(8).33 are
k and l where k has order 2, l is in class
9K/L/M/K'/L'/M', kl has order 9, kll has order 9,
klll has order 6, kllll has order 18 and
klkllkllll has order 9.
These conditions distinguish classes 9K/L/M and 9K'/L'/M'.
Standard generators of U3(8):S3 are
m and n where m is in class 2B,
n is in class 3G/G', mn has order 8, mnmnn has order 9
and (mn)3mn2mnmn2mn2 has order 14.
Standard generators of [any] 3.U3(8).S3 are
preimages M and N where ...??... .
Standard generators of U3(8).32 are o and
p where o is in class 3DEF/DEF', p is in class 9EFG/EFG',
op has order 9, opp has order 9, oppp has order 12,
opppp has order 9 and opoopp has order 7.
These conditions distinguish classes 3DEF and 3DEF'.
Standard generators of U3(8).(S3 × 3) are
q and r where q is in class 2B, r is in class
9KLM/KLM', qr has order 6, qrqrr has order 3 and
qrqrrqrrrr has order 6.
These conditions distinguish classes 9KLM and 9KLM'.
< a, b | a2 = b3 =
(ab)19 = [a, b]9 =
[a, bab]3 =
(abababab-1)3ab-1ab(ab-1)3ab(ab-1)2 =
(((ab)4(ab-1)3)2ab-1)2
= 1 >.
The representations of U3(8) available are:
-
Permutations on 513 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
-
Permutations on 3648 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- a and
b as
8 x 8 matrices over GF(8).
- a and
b as
27 x 27 matrices over GF(4).
-
Dimension 56 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
- a and
b as
133 x 133 matrices over GF(3).
- a and
b as
133 x 133 matrices over GF(3).
- a and
b as
133 x 133 matrices over GF(3).
The representations of 3.U3(8) available are:
-
Permutations on 4617 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Permutations on 32832 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
-
Dimension 3a over GF(64):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
- the natural representation.
The representations of U3(8):2 available are:
-
Permutations on 513 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
-
Permutations on 3648 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
The representations of U3(8):31 available are:
-
Permutations on 513 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
-
Permutations on 3648 points:
e and
f (Meataxe),
e and
f (Meataxe binary),
e and
f (GAP).
The representations of U3(8):6 available are:
-
Permutations on 513 points:
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
-
Permutations on 3648 points:
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
- Essentially all faithful irreducibles in characteristic 2.
-
Dimension 24 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
-
Dimension 54a over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
- restricting to U3(8) as 9a+9b+9c+9d+9e+9f.
-
Dimension 54b over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
- restricting to U3(8) as 27a+27b.
-
Dimension 192 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
-
Dimension 432 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
-
Dimension 512 over GF(2):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
-
Dimension 56 over GF(3):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
-
Dimension 133 over GF(3):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
-
Dimension 266 over GF(3):
g and
h (Meataxe),
g and
h (Meataxe binary),
g and
h (GAP).
The representations of U3(8):32 available are:
-
Permutations on 513 points:
i and
j (Meataxe),
i and
j (Meataxe binary),
i and
j (GAP).
-
Permutations on 3648 points:
i and
j (Meataxe),
i and
j (Meataxe binary),
i and
j (GAP).
The representations of U3(8).33 available are:
-
Permutations on 513 points:
k and
l (Meataxe),
k and
l (Meataxe binary),
k and
l (GAP).
-
Permutations on 3648 points:
k and
l (Meataxe),
k and
l (Meataxe binary),
k and
l (GAP).
The representations of U3(8):S3 available are:
-
Permutations on 513 points:
m and
n (Meataxe),
m and
n (Meataxe binary),
m and
n (GAP).
-
Permutations on 3648 points:
m and
n (Meataxe),
m and
n (Meataxe binary),
m and
n (GAP).
The representations of U3(8).32 available are:
-
Permutations on 513 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
-
Permutations on 3648 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
The representations of U3(8).(S3 × 3) available are:
-
Permutations on 513 points:
q and
r (Meataxe),
q and
r (Meataxe binary),
q and
r (GAP).
-
Permutations on 3648 points:
q and
r (Meataxe),
q and
r (Meataxe binary),
q and
r (GAP).
- ab is in class 19A.
- (ABABABB)42 is the central element of order 3.
- (ghghghhghghh)3 is in class 8A [forced by the choice for U3(8):2].
Go to main ATLAS (version 2.0) page.
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Go to old U3(8) page - ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 21st July 2004, from a version 1 file last updated on 25th May 2000.
Last updated 21.07.04 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.