ATLAS: Thompson group Th
Order = 90745943887872000 = 2^{15}.3^{10}.5^{3}.7^{2}.13.19.31.
Mult = 1.
Out = 1.
The following information is available for Th:
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 nontrivial 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.
Finding generators
To find standard generators for Th:

Find any element x of order 2 (by taking a suitable power of any element
of even order).
[The probability of success at each attempt is 18731 in 32768 (about 1 in 2).]

Find any element of order 21 or 39. This powers up to a 3Aelement, y, say.
[The probability of success at each attempt is 9 in 91 (about 1 in 10).]

Find a conjugate a of x and a conjugate b of y such that ab has order 19.
[The probability of success at each attempt is 192 in 14725 (about 1 in 77).]

Now a and b are 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:
 Check o(x) = 2.
 Check o(y) = 3.
 Check o(xy) = 19.
 Let z = (xy)^{3}y.
 Check o(z) = 21.
 Let w = xyy(xy)^{4}(xyy)^{2}(xy)^{2}(xyy)^{5}(xy)^{3}.
 Check o(y(z^{7})^{w}) = 2.
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:

Dimension 248 over GF(2):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 248 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 3875 over GF(3):
a and
b (Meataxe).

Dimension 248 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 248 over GF(7):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 248 over GF(13):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 248 over GF(19):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 248 over GF(31):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP).

Dimension 248 over Q (indeed, over Z[½]):
a and b (Magma).
The representation of Th available on Type II standard generators is:

Dimension 248 over GF(2):
a, b, c, d,
e, s, t, u (GAP),
a, b, c, d,
e, s, t, u (Magma).
The maximal subgroups of Th are:

^{3}D_{4}(2):3,
with (nonstandard) generators
(ab)^{8}a(ab)^{8},
(abababb(ab)^{4}abbababb)^{5},
and (nonstandard) generators
a,
ababab^{2}ab^{2}ab^{2}abababab^{2}ab^{2}abab^{2}.

2^{5}.L_{5}(2),
with standard generators a, ((abbab)^{2})^((ab)^{15}(abb)^{9}(ab)^{12}(abb)^{16}(ab)^{17}).

2^{1+8}.A_{9},
with (nonstandard) generators here.

U_{3}(8):6,
with standard generators a, (abb)^{1}babb
(same as a, bababb).

(3 × G_{2}(3)):2, with
(2, 4, 39)generators here,
mapping onto standard generators of G_{2}(3):2.

3.3^{2}.3.(3 × 3^{2}).3^{2}:2S_{4}, with generators
a^((ab)^{8}(abb)^{6}),
((ababababbababbabb)^{3})^((abb)^{13}(ab)^{5}).

3^{2}.3^{3}.3^{2}.3^{2}:2S_{4}, with generators
a^((abb)^{3}(ab)^{11}),
((abababb)^{3})^((ab)^{14}(abb)^{6}).

3^{5}:2.S_{6}, with generators
a^((abb)^{4}(ab)^{4}(abb)^{14}),
(ababababbabb)^((abb)^{12}(ab)^{4}).

5^{1+2}:4S_{4}, with generators
(((ab)^{4}(ababb)^{2}abb)^{10})^((ab)^{8}(abb)^{5}(ab)^{7}),
((abababababb)^{5})^((ab)^{16}(abb)^{11}).

5^{2}:GL_{2}(5), with generators
a^((ab)^{3}(abb)^{15}),
((abababababb)^{5})^((abb)^{8}(ab)^{9}).

7^{2}:(3 × 2S_{4}), with generators
b^((ab)^{5}(abb)^{8}(ab)^{2}),
((abababababb)^{5})^((ab)^{10}(abb)^{9}).

L_{2}(19):2,
with nonstandard generators
(ab)^{2}bab,
(abb)^{6}(abababb)^{6}(abb)^{6}.

L_{3}(3),
with (nonstandard) generators
(abb)^{8}a(abb)^{8},
(ab)^{8}(abababb)^{6}(ab)^{8}.

M_{10}, with generators
a^((ab)^{4}(abb)^{4}),
((abababababb)^{5})^((abb)^{17}(ab)^{6}).
Standard generators may be obtained from these generators (x, y) as
x, yxyy. The programs have been
combined here.

F_{465} = 31:15,
with generators here and
here. Magma versions are
here and
here.
This subgroup is
generated by (ab)^5(abb)^2ab(abb)^4, of order 31, together with an element of
order 15 that conjugates it to its 9th power. Both programs produce the same elements of Th, but the W2version is much shorter.

S_{5}, with standard generators
here.
The class representatives of the 48 conjugacy classes of Th are as follows:
 1A: identity or a^{2}.
 2A: a.^{ }
 3A: b.^{ }
 3B: (ababab^{2})^{6} or [a, bab]^{3}.
 3C: [a, babab]^{2}.
 4A: ababab^{2}ababab^{2}abab^{2}.
 4B: (abababab^{2}ab^{2})^{3}.
 5A: [a, b]^{2} or (abab^{2})^{2}.
 6A: [a, babab] or (ab)^{3}(ab^{2})^{3}.
 6B: (ab)^{6}(ab^{2})^{3}.
 6C: (ababab^{2})^{3}.
 7A: (ab)^{6}(ab^{2})^{6} or (abababab^{2})^{3}.
 8A: (ab)^{7}ab^{2} or ((ab)^{3}ab^{2}ab(ab^{2})^{2})^{3}.
 8B: ababababab^{2}(abab^{2}ab^{2}ab^{2})^{2} or (ab)^{9}(ab^{2})^{2}ab(ab^{2})^{3} or ((ab)^{7}ab^{2}ab(ab^{2})^{2})^{3}.
 9A: abab(abab^{2})^{3}ab^{2}ab^{2}.
 9B: ababababab^{2}ab^{2}abab^{2}ab^{2} or ((ab)^{5}ab^{2})^{3}.
 9C: (ababab^{2})^{2} or [a, bab].
 10A: [a, b] or abab^{2}.
 12A/B: abababab^{2}ababab^{2}ab^{2}.
 12C: (abababab^{2}ab^{2})^{2}ab^{2}.
 12D: abababab^{2}ab^{2}.
 13A: (ab)^{9}(ab^{2})^{3} or ab(abababab^{2})^{2}.
 14A: (ab)^{4}(ab^{2})^{3}.
 15A/B: (ab)^{6}ab^{2}abab^{2}ab^{2}.
 18A: (ab)^{10}(ab^{2})^{4} or abababab^{2}abab^{2}ab^{2}abab^{2} or (ab)^{5}(ababab^{2})^{2}.
 18B: ababab^{2}.
 19A: ab.^{ }
 20A: (ab)^{4}ab^{2}.
 21A: abababab^{2}.
 24A/B: (ab)^{3}ab^{2}ab(ab^{2})^{2}.
 24C/D: (ab)^{7}ab^{2}ab(ab^{2})^{2}.
 27A: (ab)^{5}ab^{2}.
 27B/C: (ab)^{7}ab^{2}abab(ab^{2})^{3} or ab(abababab^{2})^{2}abab^{2}.
 28A: (ab)^{6}ab^{2}.
 30A/B: (ab)^{5}ab^{2}abab^{2}ab^{2}.
 31A/B: (ab)^{5}(ab^{2})^{2}ab(ab^{2})^{4} or ab(ababab^{2})^{3}ab^{2}.
 36A: (ab)^{8}(ab^{2})^{2}.
 36B/C: ab(abababab^{2})^{2}ab^{2}ab^{2}.
 39A/B: (ab)^{8}ab^{2}ab(ab^{2})^{2}.
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.
An amalgam:
 Words to generate an amalgam of N(3B) and N(3B^{2}) over a common
subgroup of index 4 (in both groups) are given
here (Magma format).
The copy of N(3B) is the same as that given the maximal subgroups section; the
copy of N(3B^{2}) is a different representative.
A couple of nonmaximal subgroups with few prime divisors.

5^{1+2}:4^{2}:2, of order 4000 and index 3 in a
conjugate of Max9, with generators
a^((ab)^{5}(abb)^{14}),
((abababababb)^{5})^((abb)^{9}(ab)^{6}).
The class distribution of this subgroup in Th is (1A_{1},
2A_{175}, 4B_{1000}, 5A_{124}, 8B_{1000},
10A_{700}, 20A_{1000}).

5^{1+2}:D_{8}, of order 1000, with generators
a^((ab)^{5}(abb)^{2}),
((abababababb)^{5})^((abb)^{17}(ab)^{8}).
This subgroup has no normal 5^{2}, so does not lie in a conjugate of Max10.
The class distribution of this subgroup in Th is (1A_{1},
2A_{125}, 4B_{50}, 5A_{124}, 10A_{500},
20A_{200}).
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old Th page  ATLAS version 1.
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.