ATLAS: Higman-Sims group HS

Order = 44352000 = 29.32.53.7.11.
Mult = 2.
Out = 2.

The following information is available for HS:


Standard generators

Standard generators of the Higman-Sims group HS are a and b where a is in class 2A, b is in class 5A and ab has order 11.
Standard generators of the double cover 2.HS are preimages A and B where B has order 5 and AB has order 11.

Standard generators of the automorphism group HS:2 are c and d where c is in class 2C, d is in class 5C and cd has order 30.
Standard generators of either 2.HS.2 are preimages C and D where D has order 5.

A pair generators conjugate to a, b can be obtained as
a' = (cd)^{-1}(cdd)^{10}cd, b' = (cdcdd)^{-4}(cdd)^4(cdcdd)^4.


Black box algorithms

Finding generators

To find standard generators for HS:

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

To find standard generators for HS.2:

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

Checking generators

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

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

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

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

Presentations

Presentations of HS and HS:2 in terms of their standard generators are given below.

< a, b | a2 = b5 = (ab)11 = (ab2)10 = [a, b]5 = [a, b2]6 = [a, bab]3 = ababab2ab-1ab-2ab-1ab2abab(ab-2)4 = ab(ab2(ab-2)2)2ab2abab2(ab-1ab2)2 = abab(ab2)2ab(ab-1)2ab(ab2)2ababab-2ab-1ab-2 = 1 >.

< c, d | c2 = d5 = [c, d]3 = [c, d2]4 = ((cd)4cd-2cd-1cd-1cd-2)2 = [c, dcdcd2cd-2(cd2)2cd-1] = [c, dcdcd-2cd-1(cd-2)3] = [c, d-1cd-1cd2cd(cd2)3] = 1 >.

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


Representations

The representations of HS available are: The representations of 2.HS available are: The representations of HS:2 available are: The representations of 2.HS:2 available are:

Maximal subgroups

The maximal subgroups of the Higman-Sims group are as follows. Words provided by Suleiman and Wilson. The maximal subgroups of HS:2 are as follows.

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 HS page Go to old HS page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 16th June 2000.
Last updated 6.1.05 by SJN.
Information checked to Level 0 on 16.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.