ATLAS: Unitary group U4(2), Symplectic group S4(3)

Order = 25920 = 26.34.5.
Mult = 2.
Out = 2.

Standard generators

Standard generators of U4(2) = S4(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.U4(2) = Sp4(3) are preimages A and B where B has order 5 and AB has order 9.

Standard generators of U4(2):2 = S4(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.U4(2):2 are preimages C and D where D has order 9.


An outer automorphism of U4(2) of order 2 may be obtained by mapping (a, b) to (a, bbbb).


Presentations of U4(2) and U4(2):2 in terms of their standard generators are given below.

< a, b | a2 = b5 = (ab)9 = [a, b]3 = [a, bab]2 = 1 >.

< c, d | c2 = d9 = (cd2)8 = [c, d2]2 = [c, d3cd3] = 1 >.


The representations of U4(2) available are: The representations of 2.U4(2) available are: The representations of U4(2):2 available are:
NB [24/11/99]: We have not yet rigorously checked that all representations of U4(2):2 and 2.U4(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.] The representations of 2.U4(2):2 (ATLAS­version) available are:

Maximal subgroups

The maximal subgroups of U4(2) are as follows. The maximal subgroups of U4(2):2 are as follows.
Last updated 25.05.00,
R.A.Wilson, R.A.Parker and J.N.Bray.