We consider the orientation-preserving actions of finite groups G on pairs
(S3,Γ), where Γ is a connected graph of genus g>1, embedded
in S3. For each g we give the maximum order mg of such G acting on
(S3,Γ) for all such Γ⊂S3. Indeed we will classify all
graphs Γ⊂S3 which realize these mg in different levels: as
abstract graphs and as spatial graphs, as well as their group actions.
Such maximum orders without the condition "orientation-preserving" are also
addressed.Comment: 34 pages, to appear in Discrete Comput. Geo