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Possible facets here are the 12 pentagrams, 20 triangles, and 12 decagrams of the comodore itself together with the 30 internal squares.
The general naming code here is accordingly `gaddid-#{5/2}-#{3}-#{10/3}-#{4}`.
This gives rise for exactly 30 edge-facetings with `n`-gonal axial rotation symmetries, provided `n > 2`.
In fact, 3 have full icosahedral symmetry, 10 have 5-fold pyramidal symmetry, 3 have 5-fold antiprismatic symmetry,
11 have 3-fold pyramidal symmetry, and 3 have 3-fold antiprismatic symmetry.

gaddid-12-20-12-0 = gaddid |
gaddid-12-20-0-30 = qrid |
gaddid-0-0-12-30 = gird |
gaddid-1-5-1-5 = rastacu | gaddid-1-5-11-25 | gaddid-5-5-5-15 | gaddid-5-5-7-15 | gaddid-6-10-6-10-a |

icosahedral | 5-fold pyramidal ... | ||||||

gaddid-6-10-6-10-b | gaddid-6-10-6-20 | gaddid-7-15-5-15 | gaddid-11-15-1-25 | gaddid-11-15-11-5 | gaddid-2-10-10-20 | gaddid-10-10-2-20 | gaddid-10-10-10-10 |

... 5-fold pyramidal | 5-fold antiprismatic | ||||||

gaddid-3-7-3-9 | gaddid-3-7-9-21 | gaddid-3-15-9-15 | gaddid-6-10-6-12-a | gaddid-6-10-6-12-b | gaddid-6-10-6-18-a | gaddid-6-10-6-18-b | gaddid-9-5-3-15 |

3-fold pyramidal ... | |||||||

gaddid-9-5-9-15 | gaddid-9-13-3-21 | gaddid-9-13-9-9 | gaddid-6-6-6-12 | gaddid-6-6-6-18 | gaddid-6-14-6-12 | ||

... 3-fold pyramidal | 3-fold antiprismatic |

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