  The variety of quasigroups satisfying the identity $(xy)(zy)=xz$ mirrors the
variety of groups, and offers a new look at groups and their multiplication
tables. Such quasigroups are constructed from a group using right division
instead of multiplication. Their multiplication tables consist of circulant
blocks which have additional symmetries and have a concise presentation.
  These tables are a reincarnation of the group matrices which Frobenius used
to give the first account of group representation theory. Our results imply
that every group matrix may be written as a block circulant matrix and that
this result leads to partial diagonalization of group matrices, which are
present in modern applied mathematics. We also discuss right division in loops
with the antiautomorphic inverse property.
