There was a free plugin that made simple geodesics or Archimedian solids, I can’t remember which. I had it installed 5 years ago, but never used it. I took it out, and I can’t remember the maker or exact name.
You can google Archimedian solid angles (all side lengths will be equal) and construct a sphere, but it’s tough to locate the points in 3d space. Especially when it gets to breaking down the original solid to a second or third order structure, where everything gets kerflooey as you adjust strut lengths to maintain the optimally strong spherical placement of the vertices.
Sketchup itself will do you no favors here. It doesn’t draw circles, it draws polygons, and you’ll be working with polygons that are very irregular and difficult to predict, depending on where you slice the dome. You can’t scribe two arcs to locate a point on a plane, or make three spheres to locate a point in space accurately. To work around this, you’ll first have to draw a line at the proper 3d angle from origin, then go out the proper distance from each line. Why? Because you’ll never know if your two circles intersect at a vertex (farther) or midpoint (closer) of a side until you draw them. Spherically, it’s even tougher, and I"m not even sure it’s possible to get accuracy by intersecting SU’s “spheres”.
Everything you draw will be limited by the accuracy of your original angles, and they won’t be easy whole numbers, or particularly easy to figure out. And in 3d space, there are two of them for every point. It’s a lot of cumbersome, in other words.
I’m sure there is a workaround, but you have to get pretty deep into it to figure it out. Maybe go into pure math to generate the points, then draw them in SU.