Rudvalis group - Knowlpedia

The Rudvalis group $\mathrm{Ru}$ is the subgroup generated by the fixed pair of $28\times28$ matrices $A,B$ in dataset RuG1-f2r28B0:

\mathrm{Ru}=\langle A,B\rangle\leq\mathrm{GL}_{28}(\mathbb F_2).

The files RuG1-f2r28B0.m1 and RuG1-f2r28B0.m2 specify $A$ and $B$ , respectively; their entries are part of this concrete definition. Here $\mathbb F_2=\mathbb Z/2\mathbb Z$ is the field with two elements, $\mathrm{GL}_{28}(\mathbb F_2)$ 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{Ru}|=145926144000=2^{14}\cdot3^3\cdot5^3\cdot7\cdot13\cdot29.

It belongs to the pariah display block of sporadic groups. The selected matrices represent $\mathrm{Ru}$ itself; the double central extension denoted $2.\mathrm{Ru}$ is a different group.

Using the concrete model

The realization acts faithfully on $\mathbb F_2^{28}$ . Its matrix degree $28$ is distinct from the order of the group. In particular, changing the field of a representation requires separate data; the 28-dimensional representations of $2.\mathrm{Ru}$ in odd characteristic do not define this same matrix group. Conditions on the orders of two unspecified generators do not replace the defining matrix files.

References

ATLAS, Rudvalis group, order heading, “Representations of Ru,” and “Representations of 2.Ru.”
ATLAS, representation RuG1-f2r28B0, “About this representation” and “Download”; exact MeatAxe text matrices A and B.
ATLAS, sporadic groups, “Pariahs.”