  This paper aims to systematically study mystic reflection groups that emerged
independently in the paper [Selecta Math. (N.S.) 14 (2009), 325-372,
arXiv:0806.0867] by the authors and in the paper [Algebr. Represent. Theory 13
(2010), 127-158, arXiv:0806.3210] by Kirkman, Kuzmanovich and Zhang. A detailed
analysis of this class of groups reveals that they are in a nontrivial
correspondence with the complex reflection groups $G(m,p,n)$. We also prove
that the group algebras of corresponding groups are isomorphic and classify all
such groups up to isomorphism.
