ATLAS: Unitary group U_{4}(2), Symplectic group S_{4}(3)
Order = 25920 = 2^{6}.3^{4}.5.
Mult = 2.
Out = 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:
- All primitive permutation permutation representations.
- a and
b as
permutations on 27 points.
- a and
b as
permutations on 36 points.
- a and
b as
permutations on 40 points - the cosets of N(3AB).
- a and
b as
permutations on 40 points - the cosets of 3^3:S4.
- a and
b as
permutations on 45 points.
- a and
b as
4 × 4 matrices over GF(4) - the natural representation as U4(2).
- All faithful irreducibles in characteristic 3.
- a and
b as
5 × 5 matrices over GF(3) - the natural representation as O5(3).
- a and
b as
10 × 10 matrices over GF(3).
- a and
b as
14 × 14 matrices over GF(3).
- a and
b as
25 × 25 matrices over GF(3).
- a and
b as
81 × 81 matrices over GF(3).
- a and
b as
5 × 5 matrices over GF(25).
- a and
b as
6 × 6 matrices over GF(5).
- a and
b as
10 × 10 matrices over GF(25).
The representations of 2.U_{4}(2) available are:
- A and
B as
permutations on 80 points.
- A and
B as
permutations on 240 points.
- All faithful irreducibles in characteristic 3.
- A and
B as
4 × 4 matrices over GF(3) - the natural representation as Sp4(3).
- A and
B as
16 × 16 matrices over GF(3).
- A and
B as
40 × 40 matrices over GF(3).
The representations of U_{4}(2):2 available are:
NB [24/11/99]: We have not yet rigorously checked that all representations of U_{4}(2):2 and 2.U_{4}(2).2 are in standard generators,
but we believe this to be the case.
[Quite a few of these representations were here prior to this date.]
- All faithful irreducibles in characteristic 2.
- c and
d as
8 × 8 matrices over GF(2).
- c and
d as
6 × 6 matrices over GF(2) - the representation as O_{6}^{-}(2).
- c and
d as
14 × 14 matrices over GF(2).
- c and
d as
40 × 40 matrices over GF(2).
- c and
d as
64 × 64 matrices over GF(2) - the Steinberg representation.
- c and
d as
5 × 5 matrices over GF(3) - the representation as O_{5}(3).
- c and
d as
permutations on 36 points.
The representations of 2.U_{4}(2):2 (ATLASversion) available are:
- C and
D as
permutations on 240 points.
- C and
D as
4 × 4 matrices over GF(3).
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.
Last updated 25.05.00,
R.A.Wilson, R.A.Parker and J.N.Bray.