Hope you are well.
I'm posting the following as a favour on behalf of a friend who is not a native speaker, she just wants to confirm her answer is indeed correct, or if it could be answered in a more efficient manner.
I have uploaded an image of her question and her answer respectivly, sorry iam unable to write it out due to my lack of mathmatic symbolic knowledge.
Any input would be greatly appreciated, Thanks in advance!
Her calculations are perfectly correct. To be honest, it could happen that there is precisely one orbit, in which case the action is called transitive.
Besides, if she has learned the orbit-stabilizer theorem, she can check that $$|{\rm orb}(1)|=\frac{|G|}{|{\rm Stab}(1)|}=\frac{6}{2}=3,$$ but the size of the underlying set $H$ is precisely $3$. That is, the orbit containing $1$ must be the whole set! This also indicates the uniqueness of orbits in this action.