I know that a compound is achiral if it contains any one of the following kind of symmetry.
1.Plane of symmetry
2.Centre of symmetry
3.Normal axis of symmetry
4.Alternate axis of symmetry
It is difficult to write, for many compounds, its mirror image and then check using visualisation that if the image and compound are same or not, so we check for these 4 symmetries in molecule. My question is that for any compound do we need to check for all the symmetry elements one by one till we get a symmetry possessed by the molecule to declare it achiral, for example, does there exist any compound which has only alternate axis of symmetry and not any other kind of symmetry, so that any one will have to check all the symmetries one by one, till he reaches alternate axis of symmetry, to declare it achiral?
I know compound which have centre of symmetry, but not plane of symmetry, but I don't know about a compound which has only alternate axis of symmetry and not any other kind of symmetry. Most of the achiral compounds that I have come across have plane of symmetry or centre of symmetry. So, do we need to check for all axis of symmetry or only centre and plane of symmetry, or any other shorter combination?