ATLAS: Unitary group U_{4}(3)
Order = 3265920 = 2^{7}.3^{6}.5.7.
Mult = 3^{2} × 4.
Out = D_{8}.
Standard generators
U_{4}(3) and covers

Standard generators of U_{4}(3) are a and b where
a has order 2, b is in class 6A, ab has order 7
and abababbababb has order 5.

Standard generators of the double cover 2.U_{4}(3) are preimages
A and B where ABBB has order 5 and ABABB has order 7.

Standard generators of the quadruple cover 4.U_{4}(3) = SU_{4}(3) are preimages
A and B where ABBB has order 5 and ABABB has order 7.
These generators were chosen so that A and B have orders 2 and 6 respectively;
this choice forces AB to have order 28.

Standard generators of the triple cover 3_{1}.U_{4}(3) are preimages
A and B where A has order 2, AB has order 7
and ABAB^{3}AB^{4} has order 5.

Standard generators of the triple cover 3_{2}.U_{4}(3) are preimages
A and B where A has order 2, AB has order 7
and ABABBB has order 7.

Standard generators of the quadruple cover 3^{2}.U_{4}(3) are preimages
A and B where A has order 2 and AB has order 7.

Standard generators of (P × Q).U_{4}(3),
where P is a 2group and Q is a 3group,
map onto standard generators of both P.U_{4}(3) and Q.U_{4}(3).
The following are images of the standard generators under certain outer automorphisms:
U_{4}(3).2_{1} and covers

Standard generators of U_{4}(3).2_{1} are
c
and d where
c is in class 2B,
d has order 9,
cd has order 14
and cdcdd has order 9.
U_{4}(3).4 and covers

Standard generators of U_{4}(3).4 are
e
and f where
e is in class 4E/E',
f has order 9,
ef has order 28
and efeff has order 6.
U_{4}(3).2_{2} and covers

Standard generators of U_{4}(3).2_{2} are g and
h where g is in class 2D, h has order 7 and
gh has order 8.
U_{4}(3).2_{3} and covers

Standard generators of U_{4}(3).2_{3} are i and
j where i is in class 2F, j has order 5,
ij has order 8, ijj has order 8, ijijj has order 7
and ijijjijjj has order 10.

Standard generators of 3_{2}.U_{4}(3).2_{3} are
preimages I and J where I has order 2 and
J has order 5.

Standard generators of 3_{2}.U_{4}(3).2_{3}' are
preimages I and J where [I has order 2 and]
J has order 5.
U4(3).D_{8} and covers

Standard generators of U_{4}(3):D_{8} are o and
p where o is in class 2DD',
p is in class 6NN' and op has order 20.
We may obtain p as p = ox, where x has order 20 and ox has order 6.

Standard generators of 3^{2}.U_{4}(3):D_{8} are
preimages O and P where ... the representations given below
are on them.

Standard generators of 2.U_{4}(3).D_{8} (containing
2.U_{4}(3)) are
preimages O and P. No further conditions are needed.
The full envelope of U4(3) is 4.3^2.U4(3).D8.
The bicyclic extensions are as follows:
together with some others that I forgot,
such as 3_2.U4(3).2_3' and the like.
Other extensions for which representations
are available are:
Representations
U4(3) and covers

The representations of U4(3) available are
 a and
b as
permutations on 112 points.
 a and
b as
permutations on 162 points.
 a and
b as
permutations on 567 points.
 a and
b as
permutations on 1296 points.
 a and
b as
20 x 20 matrices over GF(2).
 a and
b as
19 x 19 matrices over GF(3).
 a and
b as
35 x 35 matrices over GF(5).
 a and
b as
90 x 90 matrices over GF(5).
 The representations of 2.U4(3) available are
 A and
B as
120 x 120 matrices over GF(5).
 The representations of 3_1.U4(3) available are
 A and
B as
6 x 6 matrices over GF(4).
 A and
B as
21 x 21 matrices over GF(25).
 The representations of 3_2.U4(3) available are
 A and
B as
36 x 36 matrices over GF(4).
 A and
B as
36 x 36 matrices over GF(25).
 The representations of 3^2.U4(3) available are
 A and
B as
12 x 12 matrices over GF(4)  reducible.
 The representations of 6_1.U4(3) available are
 A and
B as
6 x 6 matrices over GF(25).
 The representations of 12_1.U4(3) available are
 A and
B as
84 x 84 matrices over GF(25).
 A and
B as
132 x 132 matrices over GF(25).
 The representations of 12_2.U4(3) available are
 A and
B as
36 x 36 matrices over GF(25).
 The representations of 4.U4(3) available are
 A and
B as
4 x 4 matrices over GF(9)  the natural representation.
 A and
B as
120 x 120 matrices over GF(5).
U4(3).2_1 and covers

The representations of U4(3).2_1 available are
 c and
d as
20 x 20 matrices over GF(2).
 c and
d as
140 x 140 matrices over GF(2).

The representations of 12_2.U4(3).2_1 available are
 C and
D as
72 x 72 matrices over GF(5).
U4(3).4 and covers

The representations of U4(3).4 available are
 e and
f as
20 x 20 matrices over GF(2).
U4(3).2_3 and covers
U4(3).D8 and covers

The representations of U_{4}(3):D_{8} available are:

Permutations on 112 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 primitive  on the cosets of 3^4:(A6:D8).

Permutations on 252 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 imprimitive  on the cosets of U4(2):2 × 2.

Permutations on 280 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 primitive.

Permutations on 324 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 imprimitive  on the cosets of L3(4):2^2.

Permutations on 540 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 primitive  on the cosets of (U3(3) × 4):2.

Permutations on 1134 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 imprimitive  on the cosets of 2^5:S6.

Permutations on 2835 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 primitive.

Permutations on 4536 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 primitive.

Permutations on 5184 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 imprimitive  on the cosets of S7.

Permutations on 8505 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 primitive.

Permutations on 9072 points:
o and
p (Meataxe),
o and
p (Meataxe binary),
o and
p (GAP).
 imprimitive  on the cosets of C(p^3) = A6.2^2 × 2.

The representations of 3^{2}.U_{4}(3):D_{8} available are:

Permutations on 756a points:
O and
P (Meataxe),
O and
P (Meataxe binary),
O and
P (GAP).
 on the cosets of C(O) = (U4(2) × 3):2 × 2.

Permutations on 756b points:
O and
P (Meataxe),
O and
P (Meataxe binary),
O and
P (GAP).
 on the cosets of U4(2):2 × S3.

Permutations on 972 points:
O and
P (Meataxe),
O and
P (Meataxe binary),
O and
P (GAP).
 on the cosets of 3.L3(4).2^2.

The representations of 2.U_{4}(3).D_{8} available are:
The version of 2.U_{4}(3).D_{8} we have is GO_{6}^{}(3).2, the subgroup of GL_{6}(3) fixing or
negating a nondegenerate orthogonal form of minus type.
This group seems not to have a double cover of type 4.U_{4}(3).D_{8}.

Dimension 6 over GF(3):
O and
P (Meataxe),
O and
P (Meataxe binary),
O and
P (GAP),
O and P (Magma).
Maximal subgroups
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old U4(3) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 25th May 2004, from a version 1 file last updated on 29th Octocber 1999.
Last updated 28.06.07 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.