Lyons group - Knowlpedia

The Lyons group $\mathrm{Ly}$ is the subgroup generated by the fixed pair of $111\times111$ matrices $A,B$ in dataset LyG1-f5r111B0:

\mathrm{Ly}=\langle A,B\rangle\leq\mathrm{GL}_{111}(\mathbb F_5).

The files LyG1-f5r111B0.m1 and LyG1-f5r111B0.m2 specify $A$ and $B$ , respectively; their entries are part of this concrete definition. Here $\mathbb F_5=\mathbb Z/5\mathbb Z$ is the field with five elements, $\mathrm{GL}_{111}(\mathbb F_5)$ consists of invertible matrices, and the group operation is matrix multiplication over that field. The exact generator files are linked in References.

Order and structure

This group is a nonabelian finite simple group, with

|\mathrm{Ly}|=51765179004000000=2^8\cdot3^7\cdot5^6\cdot7\cdot11\cdot31\cdot37\cdot67.

It belongs to the pariah display block of sporadic groups.

Using the concrete model

The realization acts faithfully on $\mathbb F_5^{111}$ . Its matrix degree $111$ is distinct from the order of the group. The defining data are the specified matrices; conditions on the orders of two unspecified generators do not replace those data. The notation $\mathrm{Ly}$ names one finite group and has no rank or field parameter.

References

ATLAS, Lyons group, order heading and “Representations of Ly.”
ATLAS, representation LyG1-f5r111B0, “About this representation” and “Download”; exact MeatAxe text matrices A and B.
ATLAS, sporadic groups, “Pariahs.”