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I'm having a hard time determining whether the plane with normal the middle diagonal of the cube is a symmetry of the cube.

I drew pictures but even from the pictures it's really hard to tell.

If you connect two vertices that are diagonally across from each other (the long diagonal) and reflect the cube at the plane with this normal vector lying at the point exactly half way between the two points, is this reflection a symmetry of the cube?

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If you look at the cube from the direction of this diagonal, you'll see the six vertices not on the diagonal, three above the plane and three below the plane. The reflection in the plane does not take these vertices to vertices, so it is not a symmetry.

enter image description here

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