  Axial algebras are a class of commutative non-associative algebras which have
a natural group of automorphisms, called the Miyamoto group. The motivating
example is the Griess algebra which has the Monster sporadic simple group as
its Miyamoto group. Previously, using an expansion algorithm, about 200
examples of axial algebras in the same class as the Griess algebra have been
constructed in dimensions up to about 300. In this list, we see many
reoccurring dimensions which suggests that there may be some unexpected
isomorphisms. Such isomorphisms can be found when the full automorphism groups
of the algebras are known. Hence, in this paper, we develop methods for
computing the full automorphism groups of axial algebras and apply them to a
number of examples of dimensions up to 151.
