Order 12 group with 3 generators, can I reduce to 2 generators?

I'm just getting back into group theory after studying it quite a few years ago. I ran into a seemingly-simple question as I was getting started, looking for advice.

I was looking at the dihedral group with 6 elements (D6), and made a Cayley diagram of it (see below) with two concentric triangles: one triangle on the inside with a group generator "a" of order 3, with arrows going clockwise, and then one triangle on the outside with operator a's arrows going i n the counterclockwise direction, and finally connected the vertices of each triangle with another generator "x", of order 2. Simple enough so far.

Now, I was curious what group I might obtain if I extended this Cayley diagram in a natural way, not by increasing the order of a, but instead by adding additional concentric triangles around the outside. Eventually, in order to make a valid-looking Cayley diagram, I added a total of two more concentric triangles, so I have 4 triangles total. For the new outer triangles, I repeated the pattern of connecting the vertices inside each triangle with operator a, continuing the pattern of alternating arrow directions: counterclockwise on one outer triangle, and then clockwise on the next, so that the arrow patters are alternating for each triangle.

Now, in order to connect the vertices of each triangle to the next outer triangle, I introduced one more generator, "y" of order 2, and made sure each point in the diagram has arrows with all three operators (for x and y, I don't draw arrowheads, since these are order 2).

I have x and y alternating as we move between triangles, so I think the Cayley diagram here is valid and I assume this corresponds to a group.

The group must be nonabelian since axy != xay.

I know from a Google search there are only 3 nonabelian groups of order 12, so this must be isomorphic to one of them: A4, D12, or Dic3.

My question: I constructed this using 3 generators, a, x, and y, but for these other groups I see them each presented with just 2 generators. Furthermore, I don't see an obvious isomorphism just from looking at the Cayley graph structure. When I studied group theory previously, I also recall getting stumped by this question of understanding which graph I'm looking at, when it's defined in terms of different generators.

So, what group is it that I've drawn? Is there an obvious way to see A4, D12, or Dic3 hiding in this structure?

I haven't thought about it in detail, but it looks like $$\langle a,x,y \mid a^3=x^2=y^2=1, xy=yx, xax=yay =a^{-1} \rangle,$$ which is dihedral of order $12$ (note that $axy$ has order $6$, and $G = \langle ax,y \rangle$.