O'Nan group - Knowlpedia

The O'Nan group $\mathrm{O'N}$ is the subgroup generated by the fixed pair of $154\times154$ matrices $A,B$ in dataset ONG1-f3r154B0:

\mathrm{O'N}=\langle A,B\rangle\leq\mathrm{GL}_{154}(\mathbb F_3).

The files ONG1-f3r154B0.m1 and ONG1-f3r154B0.m2 specify $A$ and $B$ , respectively; their entries are part of this concrete definition. Here $\mathbb F_3=\mathbb Z/3\mathbb Z$ is the field with three elements, $\mathrm{GL}_{154}(\mathbb F_3)$ 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{O'N}|=460815505920=2^9\cdot3^4\cdot5\cdot7^3\cdot11\cdot19\cdot31.

It belongs to the pariah display block of sporadic groups. The selected matrices represent the simple group itself; the triple central extension denoted $3.\mathrm{O'N}$ is a different group.

A sporadic subgroup

There is a maximal subgroup isomorphic to the first Janko group:

J_1\hookrightarrow\mathrm{O'N},\qquad [\mathrm{O'N}:J_1]=2624832.

This is an existence statement for an embedding, not a classification of all homomorphisms between the two groups.

Using the concrete model

The realization acts faithfully on $\mathbb F_3^{154}$ . Its matrix degree $154$ 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.

References

ATLAS, O'Nan group, order heading, “Representations of O'N,” and “Maximal subgroups of O'N,” row J1.
ATLAS, representation ONG1-f3r154B0, “About this representation” and “Download”; exact MeatAxe text matrices A and B.
ATLAS, sporadic groups, “Pariahs.”