ATLAS: Unitary group U3(4)

Order = 62400 = 26.3.52.13.
Mult = 1.
Out = 4.

The following information is available for U3(4):


Standard generators

Standard generators of U3(4) are a and b where a has order 2, b has order 3 and ab has order 13.

Standard generators of U3(4):2 are c and d where c has order 2, d has order 3, cd has order 8, cdcdd has order 13 and cdcdcdcddcdcddcdd has order 10.
NB: Of course, c is in class 2B.

Standard generators of U3(4):4 are e and f where e is in class 2A, f is in class 4B or 4B', ef has order 12 and efefffeff has order 6.
NB: These conditions distinguish between classes 4B and 4B'. Classes 4B and 4B' are the classes for which the ATLAS class 4B is proxy.
With these conditions, f is conjugate to the *2 automorphism.


Automorphisms

A generating outer automorphism of U3(4) may be obtained by mapping (a, b) to ((ab)-2a(ab)2, (ab-1)-5b(ab-1)5).
An outer automorphism of U3(4) of order 2 is given by mapping (a, b) to (a, b-1)

Presentations

Presentations of U3(4), U3(4):2 and U3(4):4 in terms of their standard generators are given below.

< a, b | a2 = b3 = (ab)13 = [a, b]5 = [a, babab]3 = 1 >.

< c, d | c2 = d3 = (cd)8 = [c, d]13 = [c, dcdcdcd-1cdcd]2 = [c, d-1cdcd]5 = 1 >.

< e, f | e2 = f4 = (ef)12 = [e, f]5 = (ef2)10 = efefef2efef2ef-1ef2efefef-1ef2ef-1ef2 = 1 >.

These presentations are available in Magma format as follows:
U3(4) on a and b, U3(4):2 on c and d and U3(4):4 on e and f.


Representations

The representations of U3(4) available are: The representations of U3(4):2 available are: The representations of U3(4):4 available are:

Maximal subgroups

The maximal subgroups of U3(4) are as follows. The maximal subgroups of U3(4):2 are as follows. The maximal subgroups of U3(4):4 are as follows.

NB: Maps between the various extensions of U3(4) have not been checked for compatibility with the class definitions (or even compatibility with each other).


Conjugacy classes

Some conjugacy classes U3(4) are as follows. Some conjugacy classes U3(4):2 are as follows. Some conjugacy classes U3(4):4 are as follows. Choices made:
Main ATLAS page Go to main ATLAS (version 2.0) page.
Classical groups page Go to classical groups page.
Old U3(4) page Go to old U3(4) page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 27th August 2004, from a version 1 file last updated on 12th April 2000.
Last updated 02.09.04 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.