  Recently an algorithm has been constructed that shows the binary icosahedral
group $\2I$ together with a $T$-like gate forms the most efficient single-qubit
universal gate set. To carry out the algorithm fault tolerantly requires a code
that implements $\ico$ transversally. However, no such code has ever been
demonstrated in the literature. We fill this void by constructing a family of
distance $d = 3$ codes that all implement $2I$ transversally. A surprising
feature of this family is that the codes can be deduced entirely from symmetry
considerations that only $2I$ affords.
