ATLAS: Lyons group Ly
Order = 51765179004000000 = 2^{8}.3^{7}.5^{6}.7.11.31.37.67.
Mult = 1.
Out = 1.
The following information is available for Ly:
Standard generators of the Lyons group Ly are a and b where
a has order 2, b is in class 5A, ab has order 14 and
abababb has order 67.
Finding generators
To find standard generators for Ly:

Find any element x of order 2 by taking a suitable power of any element of even order.

Find any element of order 20, 25 or 40. This powers up to a 5Aelement, y say.

Find a conjugate a of x and a conjugate b of y, whose product has order 14,
such that abababb has order 67.
This algorithm is available in computer readable format:
finder for Ly.
Checking generators
To check that elements x and y of Ly
are standard generators:
 Check o(x) = 2.
 Check o(y) = 5.
 Check o(xy) = 14.
 Check o(xyxyxyy) = 67.
 Let z = (xy)^{5}(xyy)^{2}.
 Check o(z) = 42.
 Let r = z^{14}.
 Let s = y^{xyxyyxyxyxyy}.
 Check o(rs) = 14.
 Check o(rsrss) = 30.
This algorithm is available in computer readable format:
checker for Ly.
The representations of Ly available are:

Dimension 2480 over GF(4):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 651 over GF(3):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 111 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).

Dimension 517 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
— kindly provided by K. Lux.

Dimension 2480 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
— kindly provided by J. Müller and M. Neunhöffer.

Dimension 2480 over GF(5):
a and
b (Meataxe),
a and
b (Meataxe binary),
a and
b (GAP),
a and b (Magma).
— kindly provided by J. Müller and M. Neunhöffer.
The maximal subgroups of Ly are:
 G_{2}(5), with standard generators
(ababbb)^{7}a(ababbb)^{−7},
(abababb)^{−25}(abababbab)^{3}(abababb)^{25}.
 3.McL:2, with standard generators
(abababb)^{15}a(abababb)^{−15},
(bababb)^{−12}(abababbab)^{3}(bababb)^{12}.
 5^{3}.L_{3}(5), with nonstandard generators
here.
 2.A_{11}, with nonstandard generators
here.

5^{1+4}:4.S_{6}, with generators
here.

3^{5}:(2 × M_{11}), with generators
here.

3^{2+4}:2.A_{5}.D_{8}, with generators here.

F_{1474} = 67:22, with generators
a^{(abababb)17(abb)21},
(ababbabbbabbb)^{(abb)16(abababb)30}.

F_{666} = 37:18, with generators
here.
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 dealt with as follows: the choice of classes
for elements of orders 21, 31, 37, 40, 42, and 67 is the one used by
Mueller, Neunhoeffer, Roehr and Wilson when determining the
irrationalities in the character tables mod 37 and 67. In some cases this means
that we only know the traces on class representatives, and not the Brauer
character values, since we are not able to calculate the canonical lifting
of eigenvalues.
The choice of classes for the elements of orders 11, 22 and 33 is made independently,
using the representation of degree 2480 in characteristic 0 (or 2, or ...).
Go to main ATLAS (version 2.0) page.
Go to sporadic groups page.
Go to old Ly page  ATLAS version 1.
Anonymous ftp access is also available.
See here for details.
Version 2.0 created on 7th June 2000.
Last updated 28.06.06 by JNB.
Information checked to
Level 0 on 07.06.00 by RAW.
R.A.Wilson, R.A.Parker and J.N.Bray.