ATLAS: Exceptional group 2F4(2)'

An Apology

We apologise to the eminent mathematician whose name is usually attached to this group for removing his name from this page and those linked to or from it. The reason is that certain web­crawlers which have been scanning these pages have misinterpreted the occurrence of this name as an indication of quite a different content on these pages from that which actually pertains.

Sorry.


Order = 17971200 = 211.33.52.13.
Mult = 1.
Out = 2.

The following information is available for 2F4(2)':


Standard generators

Standard generators of the group 2F4(2)' are a and b where a is in class 2A, b has order 3 and ab has order 13.

Standard generators of its automorphism group 2F4(2) = 2F4(2)'.2 are c and d where c is in class 2A, d is in class 4F, cd has order 12 and cdcd2cd3 has order 4.

A pair of elements automorphic to (a, b) may be obtained as a' = c, b' = (cdcdcdcd)dcddcdcddd.


Black box algorithms

To find standard generators for 2F4(2)': To find standard generators for 2F4(2) = 2F4(2)'.2:

Presentations

Presentations for 2F4(2)' and 2F4(2)'.2 (respectively) on their standard generators are given below.

< a, b | a2 = b3 = (ab)13 = [a, b]5 = [a, bab]4 = (ababababab-1)6 = 1 >.

< c, d | c2 = d4 = (cd)12 = [c, d]5 = ((cd2)3cd)4d-2 = [c, (dc)3d-1(cd-1cd)2d] = [c, dcdcd-1cd2]2 = (cd)4cd2cd(cd-1)4cd(cd2cd-1)2cdcd2cdcd(cd-1)2cdcd2 = 1 >.

These presentations are available in Magma format as 2F4(2)' on a and b and 2F4(2)'.2 on c and d.


Representations

The representations of 2F4(2)' available are: The representations of 2F4(2) = 2F4(2)'.2 available are:

Maximal subgroups

The maximal subgroups of 2F4(2)' are: The maximal subgroups of 2F4(2) = 2F4(2)'.2 are:

Conjugacy classes

Representatives of the 22 conjugacy classes 2F4(2)' are given below. A program to calculate them is given here and a program to calculate representatives of the maximal cyclic subgroups is given here.

Representatives of the 29 conjugacy classes 2F4(2) = 2F4(2)'.2 are given below.

A program to calculate them is given here and a program to calculate representatives of the maximal cyclic subgroups is given here.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Sporadic groups page Go to sporadic groups page.
Exceptional groups page Go to exceptional groups page.
Old 2F4(2)' page Go to old 2F4(2)' page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 23rd April 1999.
Last updated 03.03.03 by RAW.
Information checked to Level 0 on 28.04.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.