ATLAS: Harada-Norton group HN

Order = 273030912000000 = 214.36.56.7.11.19.
Mult = 1.
Out = 2.

The following information is available for HN:


Standard generators

Standard generators of the Harada-Norton group HN are a and b where a is in class 2A, b is in class 3B, ab has order 22 and ababb has order 5.
Standard generators of its automorphism group HN:2 are c and d where c is in class 2C, d is in class 5A and cd has order 42.

A pair of elements conjugate to (a, b) may be obtained as
a' = (cd)^{-3}(cdcdcdcddcdcddcdd)^{10}(cd)^3, b' = (cdd)^{8}(cdcdd)^5(cdd)^{10}.

The outer automorphism may be realised by mapping (a,b) to (a,(abb)^-8b(abb)^8).


Black box algorithms

Finding generators

To find standard generators for HN:

This algorithm is available in computer readable format: finder for HN.

To find standard generators for HN.2:

This algorithm is available in computer readable format: finder for HN.2.

Checking generators

To check that elements x and y of HN are standard generators:

This algorithm is available in computer readable format: checker for HN.

To check that elements x and y of HN.2 are standard generators:

This algorithm is available in computer readable format: checker for HN.2.

Representations

The representations of HN available are: The representations of HN:2 available are:

Maximal subgroups

The maximal subgroups of HN are: The maximal subgroups of HN:2 are:

Conjugacy classes

A set of generators for the maximal cyclic subgroups can be obtained by running this program on the standard generators. All conjugacy classes can therefore be obtained as suitable powers of these elements.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Sporadic groups page Go to sporadic groups page.
Old HN page Go to old HN page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 14th June 2000.
Last updated 7.1.05 by SJN.
Information checked to Level 0 on 14.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.