Oh, we could well argue that the varying designs of uniform patches in TOS denote different "action groups", and Decker was in charge of the one with the Knot symbol, Wesley led the Flower group, while Kirk and Tracey were run-of-the-mill skippers from the Arrowhead and Oblong groups, respectively.
Kirk would then be doubly out of line trying to tell Decker what to do, while Decker would be a bit out of his depth trying to command Kirk like he commanded one of his own group.
It is entirely possible that the Enterprise either didn't belong to a particular group, or more likely given that we rarely see groups in any trek except under special circumstances that the group is meant to only be brought together under specific events rather than a full time posting. (Meaning that during regular operations each ship operates independently, but when trouble arises many ships come together to form a group).
Also the number of ships isn't surprising, a carrier group for example generally only has one carrier and several smaller vessels whose purpose is simply to protect it. A battle group likewise will only have 3 or 4 main battle vessels, and then again several smaller ships to support them. Also the group shown may not have been the entire group, but only part of it.