Pentachoric symmetry

Pentachoric symmetry
Rank	4
Space	Spherical
Order	120
Info
Coxeter diagram
Related polytopes
Omnitruncate	Great disprismatopentapentachoron

Pentachoric symmetry, also known as pennic symmetry and notated as A₄, is a 4D spherical Coxeter group. It is the symmetry group of the regular pentachoron.

Subgroups

• A₄+ (maximal)
• A₃×I (maximal)
• A₃+×I
• A₂×A₁×I (maximal)
• (A₂×A₁)+×I
• A₂+×A₁×I
• Extended 5-2 step prismatic symmetry (maximal)
• 5-2 step prismatic symmetry
• Extended rotational 5-2 step prismatic symmetry
• Rotational 5-2 step prismatic symmetry
• (B₂×A₁)/2×I
• (B₂+×A₁)/2×I
• A₂×I×I
• A₂+×I×I
• K₃×I+
• K₂×I×I
• K₂+×I×I
• A₁×I×I×I
• I×I×I×I

Convex polytopes with A₄ symmetry

• Pentachoron (regular)
• Rectified pentachoron (isogonal)/Joined pentachoron (isotopic)
• Truncated pentachoron (isogonal)/Tetrakis pentachoron (isotopic)
• Pentapentachoron (isogonal)/Disphenoidal triacontachoron (isotopic)
• Small rhombated pentachoron (isogonal)/Notched triacontachoron (isotopic)
• Great rhombated pentachoron (isogonal)/Sphenoidal hexecontachoron (isotopic)
• Small disprismatopentapentachoron (isogonal)/Triangular-antitegmatic icosachoron (isotopic)
• Prismatorhombated pentachoron (isogonal)/Rhombipyramidal hexecontachoron (isotopic)
• Great disprismatopentapentachoron (isogonal)/Tetrahedral hecatonicosachoron (isotopic)