  A connected undirected graph is called \emph{geodetic} if for every pair of
vertices there is a unique shortest path connecting them. It has been
conjectured that for finite groups, the only geodetic Cayley graphs which occur
are odd cycles and complete graphs. In this article we present a series of
theoretical results which contribute to a computer search verifying this
conjecture for all groups of size up to 1024. The conjecture is also verified
theoretically for several infinite families of groups including dihedral and
some families of nilpotent groups. Two key results which enable the computer
search to reach as far as it does are: if the center of a group has even order,
then the conjecture holds (this eliminates all 2-groups from our computer
search); if a Cayley graph is geodetic then there are bounds relating the size
of the group, generating set and center (which cuts down the number of
generating sets which must be searched significantly).
