Consider the action of the full symmetric group $S_3$ on the cube $[0,2] \times [0,2] \times [0,2]$.
Classify the orbits of this action and determine their cardinalities.
My Answer: What I note is that the orbit can have six possibilities: $(j,i,k)$, $(k,i,j)$, $(i,j,k)$, $(j,k,i)$, $(k,i,j)$; of the form $(i,i,k)$ with stabilizer $2$ and orbit $3$; or of the form $(i,i,i)$ just a stabilizer.
So does this mean that the orbits of the action we're looking at can be $(0,2,0)$, $(0,0,2)$, $(2,0,0)$; and $(2,2,0)$ or $(0,2,2)$ or $(2,0,2)$?