  Let X be a coherent configuration associated with a transitive group G. In
terms of the intersection numbers of X, a necessary condition for the point
stabilizer of G to be a TI-subgroup, is established. Furthermore, under this
condition, X is determined up to isomorphism by the intersection numbers. It is
also proved that asymptotically, this condition is also sufficient. More
precisely, an arbitrary homogeneous coherent configuration satisfying this
condition is associated with a transitive group, the point stabilizer of which
is a TI-subgroup. As a byproduct of the developed theory, recent results on
pseudocyclic and quasi-thin association schemes are generalized and improved.
In particular, it is shown that any scheme of prime degree p and valency k is
associated with a transitive group, whenever p>1+6k(k-1)^2.
