ATLAS: Unitary group U_{4}(2), Symplectic group S_{4}(3)
Order = 25920 = 2^{6}.3^{4}.5.
Mult = 2.
Out = 2.
The following information is available for U_{4}(2):
Standard generators of U_{4}(2) = S_{4}(3) are a and
b, where a in class 2A and b has order 5 and ab
has order 9.
Standard generators of the double cover 2.U_{4}(2) = Sp_{4}(3)
are preimages A and B where B has order 5 and
AB has order 9.
Standard generators of U_{4}(2):2 = S_{4}(3):2 are c
and d, where c in class 2C and d has order 9 and
cd has order 10.
Standard generators of either of the double covers 2.U_{4}(2):2 are
preimages C and D where D has order 9.
An outer automorphism of U_{4}(2) of order 2 may be obtained by
mapping (a, b) to (a, bbbb).
Presentations of U_{4}(2) and U_{4}(2):2 in terms of their
standard generators are given below.
< a, b  a^{2} = b^{5} = (ab)^{9} = [a, b]^{3} = [a, bab]^{2} = 1 >.
< c, d  c^{2} = d^{9} = (cd^{2})^{8} = [c, d^{2}]^{2} = [c, d^{3}cd^{3}] = 1 >.
The representations of U_{4}(2) available are:
These representations are ordered [supposedly] with respect to ab being in class 9A.
 All primitive permutation permutation representations.

Permutations on 27 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 36 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Permutations on 40a points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the cosets of N(3AB).

Permutations on 40b points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 the cosets of 3^3:S4.

Permutations on 45 points:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Some faithful irreducibles in characteristic 2.

Dimension 4a over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 All faithful irreducibles in characteristic 3.

Dimension 5 over GF(3)  the natural representation as O5(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 14 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 25 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 81 over GF(3)  the Steinberg representation:
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Some faithful irreducibles in characteristic 5.

Dimension 5a over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 6 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 10b over GF(25):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).
 Some faithful irreducibles in characteristic 0
The representations of 2.U_{4}(2) available are:

Permutations on 80 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Permutations on 240 points:
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
 All faithful irreducibles in characteristic 3.

Dimension 4 over GF(3)  the natural representation as Sp4(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 16 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).

Dimension 40 over GF(3):
A and
B (Meataxe),
A and
B (Meataxe binary),
A and
B (GAP).
The representations of U_{4}(2):2 available are:

Permutations on 27 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 36 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Permutations on 40a points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 the cosets of N(3AB).

Permutations on 40b points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 the cosets of 3^3:(S4 × 2).

Permutations on 45 points:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).
 All faithful irreducibles in characteristic 2.

Dimension 8 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 6 over GF(2)  the representation as O_{6}^{}(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 14 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 40 over GF(2):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 64 over GF(2)  the Steinberg representation:
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 5 over GF(3)  the representation as O_{5}(3):
c and
d (Meataxe),
c and
d (Meataxe binary),
c and
d (GAP).

Dimension 6 over Z:
c and d (Magma);
c and d (Magma).
 lattices E6* and E6.

Dimension 81 over Z:
c and d (Magma).
The representations of 2.U_{4}(2):2 (ATLASversion) available are:

Permutations on 240 points:
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).

Dimension 4 over GF(3):
C and
D (Meataxe),
C and
D (Meataxe binary),
C and
D (GAP).
The maximal subgroups of U_{4}(2) are as follows.
 2^{4}:A_{5}.
 S_{6} = A_{6}:2.
 3^{1+2}:2A_{4}.
 3^{3}:S_{4}.
 2.(A_{4} × A_{4}).2.
The maximal subgroups of U_{4}(2):2 are as follows.
 U_{4}(2).
 2^{4}:S_{5}.
 S_{6} × 2.
 3^{1+2}:2S_{4}.
 3^{3}:(S_{4} × 2).
 2.(A_{4} × A_{4}).2.2.
Conjugacy classes
A set of generators for the maximal cyclic subgroups of U4(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 U4(2).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.
Go to main ATLAS (version 2.0) page.
Go to classical groups page.
Go to old U4(2) page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 25th July 2000.
Last updated 16.12.04 by JNB.
Information checked to
Level 0 on 25.07.00 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.