In the mid 1960's, Zvonomir Janko discovered a finite simple group, J₁, of order 175560. [Ja65],[Ja66] His original formulation of the group was as the unique simple group with abelian 2-Sylow subgroups and an involution centralizer isomorphic to 2 × A ₅. This first Janko group is exceptional, not only as one of the 26 sporadic finite simple groups, but also as one of only 6 pariahs (a sporadic finite simple group that does not appear as a subgroup or subquotient of the Monster). [Wi86] Other than the Matthieu groups, J₁ is the smallest sporadic group.

There exists a 266-point graph, called the Livingstone graph, whose automorphism group is J₁. We study this graph in great depth, paying particular attention to how various maximal subgroups of J ₁ act on it. In each case, an attempt is made to find a simple structure within the graph that “explains” the subgroup. Upon completing this task, we use the discussion of the subgroup 2 × A₅ to build an orbifold comprised of 1463 dodecahedra whose symmetry group is exactly J₁ (and a manifold whose symmetry group is 2 × J₁).

In addition, there is a final chapter discussing the various cyclogons of the Livingstone graph. (A cyclogon of a graph is a sequence of vertices permuted by some graph automorphism.) Basic defintions are made, propositions proved, and a notion of equivalence is given, along with an exhaustive enumeration of the equivalence classes of cyclogons of various types.