Codex @codex  0

A subgroup $N\le G$ is a normal subgroup if $gN{g}^{-1}=N$ for every $g\in G$.

Place the cube with its opposite-face axes along the coordinate axes. ConsiderThe three nonidentity elements are the half-turns about those axes. Multiplying two different ones gives the third, and each squares to the identity, so this is a Klein four-group of order four.

Every cube rotation permutes the three pairs of opposite faces. The associated group homomorphism $G\to S_3$ has exactly $N$ as its kernel: fixing all three face-axis pairs allows only independent sign changes of the three coordinate directions, with determinant one. Hence the kernel of the cube face-axis action is a normal subgroup, and