  We consider finite group-actions on closed, orientable and nonorientable
3-manifolds; such a finite group-action leaves invariant the two handlebodies
of a Heegaard splitting of M of some genus g. The maximal possible order of a
finite group-action of an orientable or nonorientable handlebody of genus g > 1
is 24(g-1), and in the present paper we characterize the 3-manifolds M and
groups G for which the maximal possible order |G| = 24(g-1) is obtained, for
some G-invariant Heegaard splitting of genus g > 1. If M is reducible then it
is obtained by doubling an action of maximal possible order 24(g-1) on a
handlebody of genus g. If M is irreducible then it is a spherical, Euclidean or
hyperbolic manifold obtained as a quotient of one of the three geometries by a
normal subgroup of finite index of a Coxeter group associated to a Coxeter
tetrahedron, or of a twisted version of such a Coxeter group.
