  Coherent configurations (CCs) are highly regular colorings of the set of
ordered pairs of a "vertex set"; each color represents a "constituent digraph."
CCs arise in the study of permutation groups, combinatorial structures such as
partially balanced designs, and the analysis of algorithms; their history goes
back to Schur in the 1930s. A CC is primitive (PCC) if all its constituent
digraphs are connected.
  We address the problem of classifying PCCs with large automorphism groups.
This project was started in Babai's 1981 paper in which he showed that only the
trivial PCC admits more than $\exp(\tilde{O}(n^{1/2}))$ automorphisms. (Here,
$n$ is the number of vertices and the $\tilde{O}$ hides polylogarithmic
factors.)
  In the present paper we classify all PCCs with more than
$\exp(\tilde{O}(n^{1/3}))$ automorphisms, making the first progress on Babai's
conjectured classification of all PCCs with more than $\exp(n^{\epsilon})$
automorphisms.
  A corollary to Babai's 1981 result solved a then 100-year-old problem on
primitive but not doubly transitive permutation groups, giving an
$\exp(\tilde{O}(n^{1/2}))$ bound on their order. In a similar vein, our result
implies an $\exp(\tilde{O}(n^{1/3}))$ upper bound on the order of such groups,
with known exceptions. This improvement of Babai's result was previously known
only through the Classification of Finite Simple Groups (Cameron, 1981), while
our proof, like Babai's, is elementary and almost purely combinatorial.
  Our analysis relies on a new combinatorial structure theory we develop for
PCCs. In particular, we demonstrate the presence of "asymptotically uniform
clique geometries" on PCCs in a certain range of the parameters.
