ATLAS: Thompson group Th

Order = 90745943887872000 = 215.310.
Mult = 1.
Out = 1.

The following information is available for Th:

Standard generators

Type I standard generators of the Thompson group Th are a and b where a has order 2, b is in class 3A and ab has order 19.

Type II standard generators of the Thompson group Th are non-trivial elements a, b, c, d, e, s, t and u that satisfy the Havas–Soicher–Wilson presentation, see below.

A program to convert Type I to Type II generators and back again is available here in Magma format.

Black box algorithms

Finding generators

To find standard generators for Th:

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

Checking generators

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

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


The Soicher (or Havas–Soicher–Wilson) presentation of Th is available in Magma format here. Matrices in GAP and Magma format satisfying this presentation are given below (in the representations section). GAP matrices in separate files are also available in version 1. Note that the original words used to make these matrices seem to have been lost; the version here has been calculated anew.

The relevant reference is:
G.Havas, L.H.Soicher and R.A.Wilson. A presentation for the Thompson sporadic simple group. Groups and Computation III (Columbus, Ohio, 1999), 193–200, Ohio State Univ. Math. Res. Inst. Publ. 8, de Gruyter, Berlin, 2001.


The representations of Th available on Type I standard generators are: The representation of Th available on Type II standard generators is:

Maximal subgroups

The maximal subgroups of Th are:

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 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.

Additional information

An amalgam:

A couple of non-maximal subgroups with few prime divisors.
Main ATLAS page Go to main ATLAS (version 2.0) page.
Sporadic groups page Go to sporadic groups page.
Old Th page Go to old Th page - ATLAS version 1.
ftp access Anonymous ftp access is also available. See here for details.

Version 2.0 created on 14th April 1999.
Last updated 31.05.06 by JNB.
Information checked to Level 1 on 11.06.99 by JNB.
R.A.Wilson, R.A.Parker and J.N.Bray.