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Finally I am starting an entry Platonic 2-group.
For the moment, all it has is the statement of Epa-Ganter 16, prop. 4.1, rephrased as the diagram
$\array{ \mathcal{G}_{uni}[i] &\longrightarrow& \mathbf{B}G_{ADE} &\longrightarrow& \mathbf{B}^3 \mathbb{Z}/{\vert G_{ADE}\vert} \\ \downarrow && \downarrow && \downarrow \\ String(SU(2)) &\longrightarrow& \mathbf{B} SU(2) &\underset{\mathbf{c}_2}{\longrightarrow}& \mathbf{B}^3 U(1) }$I added a comment on what the $i$ in your diagram means.
Itâ€™s reminiscent of my quest to find a basic 2-group for Klein 2-geometry. I was looking for copies of some group sitting on the vertices of a cube, and then wondering about symmetries.
I added a comment on what the ii in your diagram means.
Thanks! I also added that to the diagram, and harmonized the notation a little.
Idle thought in passing: can these Platonic 2-groups be lifted to the fivebrane 6-group?
It sure looks like the pattern wants to continue. But since the Platonic n-groups are, crucially, not just the pullbacks of the Whitehead stages, it is not evident if it will work.
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