Third Janko group J3

The third Janko group $J_3$ is the subgroup generated by the fixed pair of $80\times80$ matrices $A,B$ in dataset J3G1-f2r80B0:

J_3=\langle A,B\rangle\leq\mathrm{GL}_{80}(\mathbb F_2).

The files J3G1-f2r80B0.m1 and J3G1-f2r80B0.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, $\mathrm{GL}_{80}(\mathbb F_2)$ 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

|J_3|=50232960=2^7\cdot3^5\cdot5\cdot17\cdot19.

It belongs to the pariah display block of sporadic groups. The selected matrices represent $J_3$ itself; the triple central extension denoted $3.J_3$ is a different group. The first Janko group and fourth Janko group are separate catalogue objects.

Using the concrete model

The realization acts faithfully on $\mathbb F_2^{80}$ . Its matrix degree $80$ 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, Janko group J3, order heading and “Representations of J3.”
ATLAS, representation J3G1-f2r80B0, “About this representation” and “Download”; exact MeatAxe text matrices A and B.
ATLAS, sporadic groups, “Pariahs.”