Graph isomorphic to symmetric group

Question

Show why the symmetry group of the graph below is isomorphic to $S_3 \times S_2$.
$S_3$ and $S_2$ are symmetric groups and $\times$ denotes direct product.

    *----------*
   /|\         |
  | | |        |
   \|/         |
    *----------*

Answer 1

  4----x-----2
 /|\         |
a b c        z
 \|/         |
  4----y-----2

I've labelled the diagram to show the degree at the vertices.

$S_3$ acts on a,b,c.

$S_2$ acts by flipping the whole thing over swapping x and y, and it is preserving a,b,c so we have direct product.