Abstract
To a set $\B$ of 4-subsets of a set $\Omega$ of size $n$ we introduce an invariant called the `hole stabilizer' which generalises a construction of Conway, Elkies and Martin of the Mathieu group $M_{12}$ based on Loyd's `15-puzzle'. It is shown that hole stabilizers may be regarded as objects inside an objective partial group (in the sense of Chermak). We classify pairs $(\Omega,\B)$ with a trivial hole stabilizer, and determine all hole stabilizers associated to $2$-$(n,4,\lambda)$ designs with $\lambda \leq 2$.
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