How can I obtain the overall rotation of multiple selected gameobjects in the unity editor?

When you are building the hexagon with tubes, build them parallel to an axis. From your image, I think you already rotated the tubes before grouping them, thus why the rotation is already off.

Don't rotate the tubes before grouping them.

That means your ring is one single mesh? In your description you said there are 6 seperate tubes...

Well in the case you only have a single model you either have to use the same math used to calculate those angles or access the vertices of the mesh manually. However analysing the vertices can be tricly because if it's a single mesh you can not simply seperate the vertices for the 6 tubes.

Just to be more exact now, the following assumes your mesh only contains those 6 tubes. No other geometry must be part of the same mesh:

• Since you imported the mesh as FBX you have to make sure you mark it as "readable" in the importer.

• Depending on how the mesh got imported there will be either a MeshFilter or a SkinnedMeshRenderer which has the mesh reference. So use GetComponent to get access to the right component and access the Mesh through it's sharedMesh property

• Get the vertices array into a local array.

• Find the local space center point of the mesh by calulating the arithmetic mean of all vertices. If your ring is actually centered around local 0,0,0 you can of course skip this step.

• Create relative vectors from the center to each vertex. To be on the safe side we just calculate all possible combinations but ignore vertices which are too close together.

• Once we have all normals we have to make sure they point in the same principal direction. A simple dot product helps to decide if we should flip the normal or not.

• Finally just combine all the normals to get the average.

Something like this:

 var mf = GetComponent<MeshFilter>();
 var mesh = mf.sharedMesh;
 var verts = mesh.vertices;
 // calculate center
 Vector3 center = verts[0];
 for(int i = 1; i < verts.Length; i++)
     center += verts[i];
 center /= verts.Length;
 
 //get all normals
 List<Vector3> normals = new List<Vector3>();
 for(int i = 0; i < verts.Length-1; i++)
 {
     var dir = (verts[i] - center).normalized;
     for(int n = i+1; n < verts.Length; n++)
     {
         var dir2 = (verts[n] - center).normalized;
         if (Mathf.Abs(Vector3.Dot(dir, dir2)) < 0.88f)
         {
             normals .Add(Vector3.Cross(dir, dir2));
         }
     }
 }
 
 // unify  and add up the normals
 Vector3 refNormal = normals[i];
 for(int i = 1; i < normals.Count; i++)
 {
     if (Vector3.Dot(refNormal, normals[i]) < 0)
         refNormal -= normal[i];
     else
         refNormal += normal[i];
 }
 refNormal.Normalize();
 var rotation = Quaternion.LookRotation(refNormal);