I've recently heard about the classification of finite near fields (which are skew fields which satisfy only a one sided distributed law), and all of them are either fields, so-called Dickson nearfields (which seem to come from simple modifications to skew fields; I haven't actually studied any of this!), or one of seven exceptional examples.
Just like fields, near fields come with a sharply two transitive group of affine transformations, and in fact it has been proven that all finite sharply two transitive actions come from these. This leads to seven exceptional sharply two transitive group actions. This can be read about here.
Are these actions well studied or connected to other exceptional/sporadic mathematical objects? The groups involve seem related to various groups with exceptional behavior (like $PSL_2(5)$).