ATLAS: Unitary group U_{3}(8)
Order = 5515776 = 2^{9}.3^{4}.7.19.
Mult = 3.
Out = 3 × S_{3}.
The following information is available for U_{3}(8):
Standard generators of U_{3}(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.U_{3}(8) = SU_{3}(8) are
preimages A and B where A has order 2
and AB has order 19.
Standard generators of U_{3}(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.U_{3}(8):2 are
preimages C and D where CDCDD has order 19.
Standard generators of U_{3}(8):3_{1} 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.U_{3}(8):3_{1} are
preimages E and F where E has order 2 and
F has order 3.
Standard generators of U_{3}(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.U_{3}(8):6 are
preimages G and H where ...??... .
Standard generators for U_{3}(8) may be obtained from those of U_{3}(8):6 as
(ghghghhgh)^{6},
(gh)^{6}.
Standard generators of U_{3}(8):3_{2} 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.U_{3}(8).3_{2} are
preimages I and J where I has order 2 and ...??... .
Standard generators of U_{3}(8).3_{3} 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 U_{3}(8):S_{3} 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)^{3}mn^{2}mnmn^{2}mn^{2} has order 14.
Standard generators of [any] 3.U_{3}(8).S_{3} are
preimages M and N where ...??... .
Standard generators of U_{3}(8).3^{2} 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 U_{3}(8).(S_{3} × 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  a^{2} = b^{3} =
(ab)^{19} = [a, b]^{9} =
[a, bab]^{3} =
(abababab^{1})^{3}ab^{1}ab(ab^{1})^{3}ab(ab^{1})^{2} =
(((ab)^{4}(ab^{1})^{3})^{2}ab^{1})^{2}
= 1 >.
The representations of U_{3}(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.U_{3}(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 U_{3}(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 U_{3}(8):3_{1} 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 U_{3}(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 U_{3}(8):3_{2} 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 U_{3}(8).3_{3} 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 U_{3}(8):S_{3} 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 U_{3}(8).3^{2} 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 U_{3}(8).(S_{3} × 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.