  We analyze finite orbits of the natural braid group action on the character
variety of the $n$ times punctured sphere. Building on recent results relating
middle convolution and finite complex reflection groups, our work implements
Katz's middle convolution to explicitly classify finite orbits in the
$SL_2(\C)$-character variety of the punctured sphere. We provide theoretical
results on the existence of finite orbits arising from the imprimitive finite
complex reflection groups and formulas for constructing such examples when they
exist. In the primitive finite complex reflection groups, we perform an
exhaustive search and provide computational results. Our contributions also
include Magma computer code for middle convolution and for computing the orbit
under this action when it is known to be finite.
