Harada–Norton group

The Harada–Norton group $\mathrm{HN}$ is the subgroup generated by the fixed pair of $133\times133$ matrices $A,B$ in ATLAS dataset HNG1-f5r133B0:

\mathrm{HN}=\langle A,B\rangle\leq\mathrm{GL}_{133}(\mathbb F_{5}).

The files HNG1-f5r133B0.m1 and HNG1-f5r133B0.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 5 elements, $\mathrm{GL}_{133}(\mathbb F_{5})$ 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 is a nonabelian finite simple group, with

|\mathrm{HN}|=273030912000000=2^{14}\cdot3^{6}\cdot5^{6}\cdot7\cdot11\cdot19.

It belongs to the Monster display block of the sporadic groups. This grouping records the ATLAS organization, without asserting a direct embedding into another displayed group.

The concrete matrix model

This gives a faithful action on $\mathbb F_{5}^{133}$ . The matrix degree $133$ is distinct from the number of group elements. The two specified matrices determine the group; tests on the orders of unspecified generators would not constitute this definition.

References

ATLAS, Harada–Norton group, order heading and “Representations.”
ATLAS, matrix model HNG1-f5r133B0, “About this representation” and “Download”; exact matrices A and B.