There are operations that are not rotations?

(I’m doing the first steps in group theory, so don’t be harsh)

Since reflections in a plane are rotations around some 3D axis, the question is: if the number of dimensions is high enough, there are group operations in the plane that are not reducible to a combination of rotations on some high enough dimension?

For example, there is some arbitrary permutation of elements that cannot be reduced to (many) rotations on some n-dimensional space?.