  A $2-(n,4,\lambda)$ design $(\Omega, \mathcal{B})$ is said to be supersimple
if distinct lines intersect in at most two points. From such a design, one can
construct a certain subset of Sym$(\Omega)$ called a "Conway groupoid". The
construction generalizes Conway's construction of the groupoid $M_{13}$. It
turns out that several infinite families of groupoids arise in this way, some
associated with 3-transposition groups, which have two additional properties.
Firstly the set of collinear point-triples forms a regular two-graph, and
secondly the symmetric difference of two intersecting lines is again a line. In
this paper, we show each of these properties corresponds to a group-theoretic
property on the groupoid and we classify the Conway groupoids and the
supersimple designs for which both of these two additional properties hold.
