Kernel of the cube face-axis action ($\ker (G\to S_3)\cong C_2\times C_2$)

Permuting the three pairs of opposite faces gives a group homomorphism from the rotational symmetry group of a cube to $S_3$. Its kernel consists of the identity and the half-turns about the three coordinate axes: these are precisely the diagonal sign matrices with determinant one. Thus it is a normal subgroup of order four, a Klein four-group.