I want to carefully trace the entire sequence of interactions between the structures when getting the symmetry group of the cube. I'm just learning how to work with structures and diagrams, so please check the course of my reasoning.

So $\mathbb{E}^3\in Ob(Top)$ -- euclidean space. $K_3 \subset \mathbb{E}^3$ -- $3$-dim cube. $V$ -- group of permutations on the vertex-set. $Sym K_3 = \{\phi\in Aut(\mathbb{E}^3 \mid \phi(K_3) = K_3\}$. All objects in the following diagram are $Mon$-category objects. $Hom(\mathbb{E}^3,\mathbb{E}^3)$ -- a monoid, because not all endomorphisms are invertible. (this is not monoidal-category, just category where all $Ob$ -- monoids).

$$\begin{array} A Sym(K_3) & \stackrel{f}{\hookrightarrow}& Aut(\mathbb{E}^3)& \stackrel{g}{\hookrightarrow} & Hom(\mathbb{E}^3,\mathbb{E}^3) \\ \cong{H} \\ V & \stackrel{d}{\hookrightarrow} & S_8 \end{array} $$

So $H$ -- isomorphism between $Sym(K_3)$, where elements are continuous functions, and $V$. I think, that $H$ -- functor $H: Mon \rightarrow Grp$. $H$ translates those monoids, which are also groups, into these same groups.

Group $V$ action on $F_{Set}(\{vertices\})$:

$$\begin{array} A \mathbb{E}^3 & \hookleftarrow & \{vertices\}\\ \downarrow{F_{Set}} & &\downarrow{F_{Set}}\\ X & \hookleftarrow & F_{Set}(\{vertices\}) & \xleftarrow[\phi]{pr_{ F_{Set}(\{verteces\})}} & F_{Set}(\{vertices\})\times Y & \xrightarrow{pr_Y} & Y & \hookrightarrow &S\\ & & & & & &\uparrow{G_{Set}}&& \uparrow{G_{Set}}\\ & & & & & & V &\hookrightarrow & S_8 \end{array} $$

Where $F_{Set}$ and $G_{Set}$ -- forgetful functors, $F_{Set}$ from $Top \rightarrow Set$ and $G_{Set}$ from $Grp\rightarrow Set$, $Y$ -- group $V$ structure carrier. $\phi$ -- action of group $V$ on set $X$ $\phi(g, x) = g \cdot x$, $\{vertices\}$ -- vertices (discrete subspace) of cube in euclidean space $\mathbb{E}^3$.

So I want to proof, that $|Sym(K_3)| = |Vx||Vx,e||V_{x,e}| = 48$. Where $x$ -- one of vertices -- point in $\mathbb{E}^3$, and $e$ -- edge exiting the vertex. Obviously, a vertex orbit consists of eight vertices: $|Vx| = 8$, from each vertex there are three edges, therefore the orbit of each of them $|Vx,e| = 3$ so $\Rightarrow$ $|V_{x,e}| = 2$ -- $id$ and reflection relative to the plane passing through the edge.

And I do not really understand how to interpret the edge here? As a subspace of $\mathbb{E}^3$ or like a pair of vertices $(x_1,x_2)$?

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    $\begingroup$ Why this question has -1 rep? Where I am incorrect? $\endgroup$ – Just do it Dec 8 '18 at 2:24

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