Fourth Janko group J4

The fourth Janko group $J_4$ is the subgroup generated by the fixed pair of $112\times112$ matrices $A,B$ in dataset J4G1-f2r112B0:

J_4=\langle A,B\rangle\leq\mathrm{GL}_{112}(\mathbb F_2).

The files J4G1-f2r112B0.m1 and J4G1-f2r112B0.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, $\mathrm{GL}_{112}(\mathbb F_2)$ 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_4|=86775571046077562880 =2^{21}\cdot3^3\cdot5\cdot7\cdot11^3\cdot23\cdot29\cdot31\cdot37\cdot43.

It belongs to the pariah display block of sporadic groups. The first Janko group and third Janko group are different objects, with different orders.

Using the concrete model

The realization acts faithfully on $\mathbb F_2^{112}$ . Its matrix degree $112$ is distinct from the order of the group. The fixed pair uses the type I generator convention. The defining data are the specified matrices; conditions on the orders of two unspecified generators do not replace those data.

References

ATLAS, Janko group J4, order heading and “Representations of J4.”
ATLAS, representation J4G1-f2r112B0, “About this representation” and “Download”; exact MeatAxe text matrices A and B.
ATLAS, sporadic groups, “Pariahs.”