Problem #WSP-000351

Problem

Imagine a cube that’s turquoise on the front, pink on top, yellow on the right, white on left, dark blue on back and orange on the bottom. If Arne rotates this \(180^{\circ}\) about the line through the middles of the turquoise and dark blue sides, then does it again, he gets back to the original cube. If Arne rotates this \(90^{\circ}\) about that same line, then does that three more times, then he also gets back to the original cube.
Is there a rotation he could do, and then do twice more, to get back to the original cube?