  Majorana theory was introduced by A. A. Ivanov as an axiomatic framework in
which to study objects related to the Monster simple group and the Griess
algebra. Since its inception, it has been used to construct a number of new and
important subalgebras of the Griess algebra. The objects at the centre of the
theory are known as Majorana algebras and can be studied either in their own
right, or as Majorana representations of finite groups.
  In this paper, we present an algorithm to construct the Majorana
representations of a given group. We also list a number of groups for which we
have constructed Majorana representations, including some which give new
examples of Majorana algebras. This work is inspired by that of A. Seress.
