Kernel of the cube face-axis action
= Kernel of the cube face-axis action
{title2=$\ker(G\to S_3)\cong C_2\times C_2$}
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>.