Hall–Janko group J2

The Hall–Janko group $J_2$ , also written $\mathrm{HJ}$ , is the group with presentation

J_2=\langle a,b\mid a^2=b^3=(ab)^7=[a,b]^{12}=w^6=1\rangle, \qquad w=ababab^{-1}abab^{-1},

where the commutator convention is $[a,b]=a^{-1}b^{-1}ab$ . These are defining relations for the group, not just conditions for recognizing generators in an already known group.

Order and a permutation model

This is a nonabelian simple group of order

|J_2|=604\,800=2^7\cdot3^3\cdot5^2\cdot7.

A faithful action on 100 points is specified by the pair J2G1-p100B0.g1, J2G1-p100B0.g2. A point stabilizer has three orbits of sizes $1,36,63$ .

Geometry and neighboring groups

The Hall–Janko graph has 100 vertices and valency 36. Its full automorphism group is $J_2:2$ ; $J_2$ is the simple subgroup of index two. The graph is not uniquely specified by its strongly regular parameters alone.

The group $J_2$ embeds in the sporadic Suzuki group: $\mathrm{Suz}$ has a maximal subgroup $J_2:2$ , containing $J_2$ with index two. Thus the embedded $J_2$ itself is not maximal there.

References

ATLAS: J2, order and presentations; full presentation J2G1-P2, with Magma relator file.
Representation J2G1-p100B0, “About this representation”; exact GAP data generator a and generator b.
A. E. Brouwer, Hall–Janko graph, opening paragraph, “Group,” and “Construction.”
ATLAS: Suz, “Maximal subgroups of Suz,” J2:2 row.