Can we represent an element of $G=C_3\rtimes C_2$ as $(a,b)$ like we do in the direct product?
Because when I draw a Cayley diagram of $G$, I don't know how to label each node and arrow without the elements of $S_3$ or $D_3$.
In Carter's Visual Group Theory, he constructs Cayley diagrams for such semi-direct products and moves on without specifying how do we name each generator, or arrow and node.
The problem is what he calls 'rewiring', which changes the direction of an arrow or connected nodes so that the diagram describes an automorphism of the original diagram. But when we do that, the reversed or changed arrow doesn't represent what they used to.
For example, the usual Cayley diagram of $C_3$ is $0\to 1\to 2\to 0$. Here the arrow represents $+1$, or just $1$.
And the rewiring of it is $0\to 2\to 1\to 0$. Here the arrow represents $-1$, or $+2$, or just $2$.
They are two different arrows, so must be distinguished in the Cayley diagram, I guess? But then we lose the isomorphism with $S_3$ and $D_3$.
So, the question is, why the same arrows don't function as the same?