# Motivation for the term “transitive” group action

I have two questions:

1. In a text, I read that a group permutes pairs of faces of a solid transitively. Geometrically, what are they referring to, and what is an example of when a group may not permute some aspect of a geometric object transitively?
2. The definition I usually find for transitive is that there is only one orbit, or that $Gx=Gy$ for all $x,y$. How does this relate to the usual notion of transitivity (like, in equivalence relations?)
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