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Possible facets here are the 20 triangles of the comodore itself together with the 12 internal pentagons.
The general naming code here is accordingly `ike-#{3}-#{5}`.
This gives rise for exactly 13 edge-facetings with `n`-gonal axial rotation symmetries, where `n > 2`.
In fact, 2 have full icosahedral symmetry, 5 have 5-fold pyramidal symmetry, 1 has 5-fold antiprismatic symmetry,
3 have 3-fold pyramidal symmetry, and 2 have 3-fold antiprismatic symmetry.

A well-known edge-faceting with just digonal symmetry (i.e. mirror symmetry) is the Johnson solid J62 (mibdi).

ike-20-0 = ike |
ike-0-12 = gad |
ike-5-3 = teddi | ike-7-3 | ike-13-3 | ike-6-6-a | ike-6-6-b |

icosahedral | 3-fold pyramidal | 3-fold antiprismatic | ||||

ike-5-1 = peppy | ike-5-5 | ike-5-7 | ike-10-6 |
ike-15-1 = gyepip |
ike-10-2 = pap | |

5-fold pyramidal | 5-fold antipr. |

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