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Possible facets here are the 20 triangles, 12 decagrams, and 12 pentagons of the comodore itself together with the 20 internal hexagons.
The general naming code here is accordingly `gidditdid-#{3}-#{10/3}-#{5}-#{6}`.
This gives rise for exactly 15 edge-facetings with `n`-gonal axial rotation symmetries, where `n > 2`.
In fact, 3 have full icosahedral symmetry, 6 have 5-fold pyramidal symmetry, and 6 have 3-fold pyramidal symmetry.

gidditdid-20-12-12-0 = gidditdid |
gidditdid-20-0-12-20 = giid |
gidditdid-0-12-0-20 = giddy | |||

icosahedral | |||||

gidditdid-10-2-6-10 | gidditdid-10-6-2-10 | gidditdid-10-6-6-10-a | gidditdid-10-6-6-10-b | gidditdid-10-6-10-10 | gidditdid-10-10-6-10 |

5-fold pyramidal | |||||

gidditdid-6-6-6-10 | gidditdid-10-6-6-6 | gidditdid-10-6-6-10-c | gidditdid-10-6-6-10-d | gidditdid-10-6-6-14 | gidditdid-14-6-6-10 |

3-fold pyramidal |

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