ATLAS: Unitary group U5(2)

Order = 13685760 = 210.35.5.11.
Mult = 1.
Out = 2.

The following information is available for U5(2):


Standard generators

Standard generators of U5(2) are a and b where a is in class 2A, b has order 5 and ab has order 11.
(Wlog we take ab in 11A.)

Standard generators of U5(2).2 are c and d where c has order 2 (so is in class 2C), d has order 4 (so is in class 4D), cd has order 11 and cdcdd has order 4.
(Wlog we take cdcdcddcdcdcdcddcdcdd in class 16B.)


Automorphisms

An outer automorphism may be obtained by mapping (a, b) to (a, (abb)^-5b(abb)^5). Another outer automorphism may be obtained by mapping (a, b) to (a, bbbb).
Both these automorphisms have order 2.

We may obtain standard generators of U5(2):2 as c = u and d = u(ab2)2(ab3)3(ab2)2, where u is the automorphism u : (a, b) -> (a, b-1).


Presentations

Presentations of U5(2) and U5(2):2 on their standard generators are given below.

< a, b | a2 = b5 = (ab)11 = [a, b]3 = [a, b2]3 = [a, bab]3 = [a, bab2]3 = 1 >.

< c, d | c2 = d4 = (cd)11 = (cd2)6 = (cdcd2)4 = [c, d]9 = (cd-1(cdcdcd-1)2cd2)2 = [d2, cdcdc]3 = 1 >.

These presentations are available in Magma format as follows: U5(2) on a and b and U5(2):2 on c and d.


Representations

The representations of U5(2) available are: The representations of U5(2):2 available are:

Maximal subgroups

The maximal subgroups of U5(2) are as follows. The maximal subgroups of U5(2):2 are as follows.

Conjugacy classes

Generators of the maximal cyclic subgroups of U5(2) are available here, and a program to obtain representatives of all conjugacy classes of U5(2) from these is available here.

Generators of the maximal cyclic subgroups of U5(2):2 are available here, and a program to obtain representatives of all conjugacy classes of U5(2):2 from these is available here.


Main ATLAS page Go to main ATLAS (version 2.0) page.
Classical groups page Go to classical groups page.
Old U5(2) page Go to old U5(2) page - ATLAS version 1.
ftp access Anonymous ftp access is also available on for.mat.bham.ac.uk.

Version 2.0 created on 14th December 2001.
Last updated 10.11.03 by JNB.
Information checked to Level 0 on 14.12.01 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.