S4:{5,5}

Statistics

genus ^c	4, orientable
Schläfli formula ^c	{5,5}
V / F / E ^c	12 / 12 / 30
notes
vertex, face multiplicity ^c	1, 1
Petrie polygons holes 2nd-order Petrie polygons	10, each with 6 edges 20, each with 3 edges 6, each with 10 edges
antipodal sets	6 of ( 2v, 2f, p2 ), 15 of ( 2e )
rotational symmetry group	A5, with 60 elements
full symmetry group	120 elements.
its presentation ^c	< r, s, t \| t², (rs)², (rt)², (st)², r^‑5, (s^‑1r)³ >
C&D number ^c	R4.6
The statistics marked ^c are from the published work of Professor Marston Conder.

Relations to other Regular Maps

It is self-dual.

Its Petrie dual is N10.5′.

It is a 2-fold cover of N5:{5,5}.

It can be 2-split to give R13.8′.

It can be rectified to give S4:{5,4}.

Its 2-hole derivative is the icosahedron.

It can be derived by stellation (with path <>/2) from the icosahedron. The density of the stellation is 3.
It can be derived by stellation (with path <1/-1>) from the icosahedron. The density of the stellation is 3.

List of regular maps in orientable genus 4.

Underlying Graph

Its skeleton is icosahedron.

Comments

This is the small stellated dodecahedron, embedded in the surface where it is at home, instead of painfully immersed in ℝ³. It is also the great dodecahedron, likewise.

Other Regular Maps

General Index