The group of orientation preserving isometries of Cube is $S_4$. But if we allow orientation reversing isometries also, then the group will be of order 48. What is this group (Structure)? ( Part of answer can be $S_4\rtimes \mathbb{Z}_2$, because $S_4\triangleleft G$, $\mathbb{Z}_2\leq G$, which contains a reflection in a plane passing through centers of four vertical faces; and so $S_4\cap \mathbb{Z}_2=1$. But again what is this semidirect product? There are three semidirect products of $S_4$ by $\mathbb{Z}_2$)
Symmetries of Cube
6
$\begingroup$
group-theory
finite-groups
1 Answers
19
One of the orientation-reversing isometries is the inversion $-I$, which commutes with all orientation-preserving isometries. So the group is just the direct product of $S_4$ with $\mathbb{Z}_2$.