Hall–Janko graph

• In U₃ there are 36 simple maximal subgroups of order 168. These are the vertices of a subgraph, the U₃ graph. A 168-subgroup has 14 maximal subgroups of order 24, isomorphic to S₄. Two 168-subgroups are called adjacent when they intersect in a 24-subgroup. The U₃ graph is strongly regular, with parameters
• There are 63 involutions. A 168-subgroup contains 21 involutions, which are defined to be neighbors.
• Outside U₃ let there be a 100th vertex C, whose neighbors are the 36 168-subgroups. A 168-subgroup then has 14 common neighbors with C and in all 1+14+21 neighbors.
• An involution is found in 12 of the 168-subgroups. C and an involution are non-adjacent, with 12 common neighbors.
• Two involutions are defined as adjacent when they generate a dihedral subgroup of order 8. An involution has 24 involutions as neighbors.