# ATLAS: Lyons group Ly

Order = 51765179004000000 = 28.37.56.7.11.31.37.67.
Mult = 1.
Out = 1.

The following information is available for Ly:

### Standard generators

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.

### Black box algorithms

#### 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 = z14
• Let s = yxyxyyxyxyxyy
• Check o(rs) = 14
This algorithm is available in computer readable format: checker for Ly.

### Representations

The representations of Ly available are:
• Dimension 2480 over GF(4): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 651 over GF(3): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 111 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
• Dimension 517 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP).
- kindly provided by K. Lux.
• Dimension 2480 over GF(5): a and b (Meataxe), a and b (Meataxe binary), a and b (GAP)
- 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)
- kindly provided by J. Müller and M. Neunhöffer.

### Maximal subgroups

The maximal subgroups of Ly 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.
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 ...).
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