  For any finite group $G$, a natural question to ask is the order of the
smallest possible automorphism group for a Cayley graph on $G$. A particular
Cayley graph whose automorphism group has this order is referred to as an MRR
(Most Rigid Representation), and its Cayley index is a numerical indicator of
this value. Study of GRRs showed that with the exception of two infinite
families and seven individual groups, every group admits a Cayley graph whose
MRR is a GRR, so that the Cayley index is 1. The full answer to the question of
finding the smallest possible Cayley index for a Cayley graph on a fixed group
was almost completed in previous work, but the precise answers for some finite
groups and one infinite family of groups were left open. We fill in the
remaining gaps to completely answer this question.
