Uniforms

These are the nine uniform polyhedra of the Klein quartic. They can be divided into the regulars (2), quasiregular (1), truncates (2), cantellate (1), omnitruncate (1) and the non-Wythoffians (2). All of these beside the non-Wythoffians have... say, Q₃ symmetry, of order 336. Sure, let's call it Q₃.

List

Regulars

• Quartic icositetrahedron (quart) — formally called the Klein map, has 24 heptagonal faces, with the Schläfli symbol {7,3}₈.
• Quartic pentecontahexahedron (klein) — formally called the dual Klein map, has 56 triangular faces, with the Schläfli symbol {3,7}₈. Has 19 diminishings, one of which is uniform.

Quasiregular

• Quartic pentecontahexicositetrahedron (kuq) — has 56 triangular and 24 heptagonal faces, being the rectification of either regular. Has three diminishings that behave like rotundae.

Truncates

These are the truncations of the regulars.

• Truncated quartic icositetrahedron (tiqua) — has 56 triangular and 24 heptagonal symmetry tetradecagonal faces.
• Truncated quartic pentecontahexahedron (tikle) — has 56 triangular symmetry hexagonal and 24 heptagonal faces.

Cantellate

• Small quartic rhombipentecontahexicositetrahedron (sirkuq) — has 56 triangular, 84 rectangular symmetry square and 24 heptagonal faces, being the cantellation of either regular. It is also the rectification of the quartic pentecontahexicositetrahedron, with it being made uniform. It has 158 diminishings and gyrations.

Omnitruncate

• Great quartic rhombipentecontahexicositetrahedron (girkuq) — has 56 triangular symmetry hexagonal, 84 rectangular symmetry square and 24 heptagonal symmetry tetradecagonal faces, being the omnitruncation of either regular. It is also the truncation of the quartic pentecontahexicositetrahedron, with it being made uniform..

Non-Wythoffian

These cannot be obtained by performing Wythoffian operations on the regulars. The first one is standard for any 3D symmetry, but the second is much more unique...

• Snub quartic pentecontahexicositetrahedron (sniq) — has 84 asymmetrical triangular, 56 chiral triangular and 24 chiral heptagonal faces, being the alternation of the omnitruncation of either regular. Has Q₃+ symmetry.
• Disantitridiminished quartic pentecontahexahedron (datiduk) — often simply called the tridiminished quartic pentecontahexahedron (tiduk), has 14 chiral triangular, 21 bilateral symmetry triangular and 3 heptagonal faces. As the name implies, it can be made from diminishing three opposite sets of vertices from the quartic pentecontahexahedron. As a result, it is kind of an analog to the pentagonal antiprism with the icosahedron. Has thrice-heptagonal symmetry of order 42. In the image, the heptagons are represented as a red group of triangles.