I want to give the standard representation of the complete octaedergroup $O_h$ in $\mathbb{R}^3$.
To which group is $O_h$ isomorphic, and how to obtain a standard representation of the group?
What are the conjugation classes of the group and what are the generators and relations in the group? I want to obtain the character table of $O_h$ and the standard representation (the last one is more important!).
Thanks :)
If I correctly guessed what $O_h$ is, I would go about it as follows. The vertices of a regular octahedron are (in a suitable coordinate system) the points $(\pm1,0,0),(0,\pm1,0),(0,0,\pm1)$. This suggests strongly that any permutation of the coordinates $xyz$ will be a symmetry. And so will be the linear mappings that simply change the sign of one or more coordinates. Therefore we have $$ O_h\cong (C_2\times C_2\times C_2)\rtimes S_3 $$ as a semidirect product, where $S_3$ acts by permuting the components of the threefold product. Alternatively we get a wreath product $O_h\cong C_2\wr S_3$.
So it is the same subgroup of order $48$ in $O_3(\mathbb{R})$ as the group of symmetries of a cube (well, we can place the vertices of a regular octahedron at the centers of the faces of a cube, so this is hardly a surprise).
Getting the character of the defining/standard representation is now easy. The permutations act by $3\times3$ permutation matrices and the sign changes act by diagonal matrices with $\pm1$s along the diagonal.