Given is the graph: enter image description here

I am interested in determining the orbit of the point 1 and also to determine the amount of symmetries that fix each of the points 1, 2 and 3?

My approach: Notice that the orbit of 1 is simply the cycle $\{ 1,2,3\}$ because we cannot invert the inner triangle and outer triangles, this would not preserve the graph. We can however cycle the smaller triangles along the central triangle (by means of a rotation over $\frac{2}{3}\pi$, around the centre).

For the second part I notice that the identity fixes all three points, then each point is fixed by the reflection that has its line of reflection through that point. This reflection would swap two triangles. Behold my paint skills below:enter image description here There are in total 3 such reflections, together with the identity we have 4 symmetries that meet the description.

Is this convincing and correct?


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