ATLAS: Unitary group U4(3)
Order = 3265920 = 27.36.5.7.
Mult = 32 × 4.
Out = D8.
Standard generators
U4(3) and covers
-
Standard generators of U4(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.U4(3) are preimages
A and B where ABBB has order 5 and ABABB has order 7.
-
Standard generators of the quadruple cover 4.U4(3) = SU4(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 31.U4(3) are preimages
A and B where A has order 2, AB has order 7
and ABAB3AB4 has order 5.
-
Standard generators of the triple cover 32.U4(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 32.U4(3) are preimages
A and B where A has order 2 and AB has order 7.
-
Standard generators of (P × Q).U4(3),
where P is a 2-group and Q is a 3-group,
map onto standard generators of both P.U4(3) and Q.U4(3).
The following are images of the standard generators under certain outer automorphisms:
U4(3).21 and covers
-
Standard generators of U4(3).21 are
c
and d where
c is in class 2B,
d has order 9,
cd has order 14
and cdcdd has order 9.
U4(3).4 and covers
-
Standard generators of U4(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.
U4(3).22 and covers
-
Standard generators of U4(3).22 are g and
h where g is in class 2D, h has order 7 and
gh has order 8.
U4(3).23 and covers
-
Standard generators of U4(3).23 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 32.U4(3).23 are
preimages I and J where I has order 2 and
J has order 5.
-
Standard generators of 32.U4(3).23' are
preimages I and J where [I has order 2 and]
J has order 5.
U4(3).D8 and covers
-
Standard generators of U4(3):D8 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 32.U4(3):D8 are
preimages O and P where ... the representations given below
are on them.
-
Standard generators of 2.U4(3).D8 (containing
2.U4(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 U4(3):D8 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 32.U4(3):D8 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.U4(3).D8 available are:
The version of 2.U4(3).D8 we have is GO6-(3).2, the subgroup of GL6(3) fixing or
negating a non-degenerate orthogonal form of minus type.
This group seems not to have a double cover of type 4.U4(3).D8.
-
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.