ATLAS: Non­split extension 214.U7(2)

Order = 3732063900024176640 = 235.38.5.7.11.43.
Mult = 1.
Out = 2.
The page for the image U7(2) is available here.

The following information is available for 214.U7(2):

Standard generators

Standard generators of 214.U7(2) are a and b where a is in class 2C, b has order 7 and ab has order 33. No extra conditions (such as abb having order 45) are required. These generators map onto standard generators of U7(2).

Standard generators of 214.U7(2):2 are not defined.

Note that 214.U7(2) has a unique conjugacy class of subgroups of index 10836, and we may take a to be a central involution in such a subgroup. In fact, this subgroup turns out to be C(a). Also, a happens to be in a class of 3­transpositions, and a commutes with 2644 of its conjugates. The orbits of C(a) [acting by conjugation] on the conjugates of a have sizes 1, 3, 2640 and 8192.

Remark: As far as we know, this group was discovered by J.I.Hall. He also discovered its 3­transposition property.

Automorphisms

An outer automorphism of 214.U7(2) may be obtained by mapping (a, b) to (a, b-1).

Presentation

A presentation of 214.U7(2) on its standard generators is given below.

< a, b | a2 = b7 = (ab)33 = [a, b]3 = [a, b2]3 = [a, b3]3 = [a, bab]2 = [a, b2ab2]2 = [a, bab2]3 = (abab3abab-2ab-2)6 = 1 >.

This presentation is available in Magma format as follows: 214.U7(2) on a and b.

Representations

The representations of 214.U7(2) available are:
Main ATLAS page Go to main ATLAS (version 2.0) page.
Miscellaneous groups page Go to miscellaneous groups page.
Old 214U7(2) page There is no old 214U7(2) page or U7(2) page in the ATLAS version 1.
ftp access Anonymous ftp access is also available on sylow.mat.bham.ac.uk.

Version 2.0 created on 10th December 1999.
Last updated 02.08.00 by JNB.
Information checked to Level 0 on 10.12.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.