ATLAS: Unitary group U3(7)

Order = 5663616 =
Mult = 1.
Out = 2.

The following information is available for U3(7):

Standard generators

Standard generators of U3(7) are a and b where a has order 2, b has order 3, ab has order 43 and ababb has order 4.

Type I standard generators of U3(7):2 are c and d where c is in class 7A, d is in class 4D and cd has order 6.
Type II standard generators of U3(7):2 are e and f where e is in class 2B, f has order 3, ef has order 8 and efeff has order 8.



An outer automorphism of U3(7) can be taken to map (a, b) to (a, bb).

Black box algorithms

To find standard generators for U3(7): To find standard generators for U3(7).2:


Presentations of U3(7) and U3(7):2 (respectively) on their standard generators are given below.

< a, b | a2 = b3 = (ab)43 = [a, b]4 = [a, bababab]3 = ab(ababab-1)2((ab)3(ab-1)3)3 = 1 >.

< c, d | c7 = d4 = (cd)6 = ... = 1 >.

< e, f | e2 = f3 = (ef)8 = [e, f]8 = (efefef-1)8 = ... = 1 >.

These presentations are available in Magma format as follows: U3(7) on a and b, U3(7):2 on c and d and U3(7):2 on e and f.


The representations of U3(7) available are: The representations of U3(7):2 available are (NONE):

Maximal subgroups

The maximal subgroups of U3(7) are: The maximal subgroups of U3(7):2 are [NOT YET FOUND]:

Conjugacy classes

The class representatives of the 48 conjugacy classes of Th are as follows: An element cannot be obtained as a power of an element of greater order just if it is in class 18B or has order at least 19.

A set of generators for the maximal cyclic subgroups can be obtained by running this program [NOT THERE YET] on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements. Problems of algebraic conjugacy are not dealt with.
Main ATLAS page Go to main ATLAS (version 2.0) page.
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Old U3(7) page Go to old U3(7) page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 13th October 2003.
Last updated 15.10.03 by JNB.
Information checked to Level 0 on 13.10.03 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.