Validity of Direct Product Structure of Symmetry Group

[Newtime]

In short, if we consider the group of symmetries of a regular octahedron, we see (or at least, the author of "Groups, Graphs and Trees" saw...) that the group is isomoprhic to Z₂$$\otimes$$Z₂$$\otimes$$Z₂$$\otimes$$S₃ - particularly since if we break up the vertices into 3 groups of front-back, top-bottom and left-right we get the first three factors and the second factor is obtained by further permutation. But my question is how does an element of a direct product act on an element of a graph? If we take a vertex v in the graph, since all symmetries commute here, if we apply a symmetry h in the direct product to v, we are thus applying 4 symmetries each in one of the factors of the direct product. So is the vertex taken to "coordinates" of whatever the direct product indicates similar to the way we consider 3 dimensional coordinates as in if we have the coordinates (1,2,3) we could move 2 in the y direction then 1 in the x and 3 in the z or 3 in the z direction THEN 2 in the y then 1 in the x etc. and thus the same process for the location of the vertex v after h is applied. Thanks in advance for your help - I know I probably rambled a bit...

 

[fresh_42]

A direct product is written with $\times$ or $oplus$ for additionally written groups. $\otimes$ notes the tensor product. Anyway. You number the vertices of the octahedron and observe how group elements permute them. The group elements are defined as the geometric transformations which turn the octahedron onto itself, e.g. an upside down transformation. Then we have several reflection planes etc. The structure is defined by the way they form another transformation by consecutive application of two. E.g. upside down applied twice is the identity. It is a candidate for one copy of $\mathbb{Z}_2$. In general it is not so easy to determine the group structure by its multiplication table. However, geometric figures are relatively simple. Let us assume all such transformations can be written as a product of three different involutions and a $3-$cycle: $T=I_1I_2I_3C$.
In case of a direct product, we get the structure by $TT'=(I_1I_2I_3C)(I'_1I'_2I'_3C')=I_1I'_1I_2I'_2I_3I'_3CC'$. So we have to check, whether this is true for all the transformations.