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Possible facets here are the 12 pentagrams, 20 triangles, and 20 hexagons of the comodore itself together with the 12 internal decagons.
The general naming code here is accordingly `siid-#{5/2}-#{3}-#{6}-#{10}`.
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.

siid-12-20-20-0 = siid |
siid-12-20-0-12 = sidditdid |
siid-0-0-20-12 = siddy | |||

icosahedral | |||||

siid-2-10-10-6 | siid-6-10-10-2 | siid-6-10-10-6-a | siid-6-10-10-6-b | siid-6-10-10-10 | siid-10-10-10-6 |

5-fold pyramidal | |||||

siid-6-6-10-6 | siid-6-10-6-6 | siid-6-10-10-6-c | siid-6-10-10-6-d | siid-6-10-14-6 | siid-6-14-10-6 |

3-fold pyramidal |

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