Polytope of Type {3,3,3} with group SmallGroup(120,34)

Overview

Group: SmallGroup(120,34)
Rank: 4
Schläfli Type: {3,3,3}
Vertices, edges, …: 5, 10, 10, 5
Order of s₀s₁s₂s₃: 5
Order of s₀s₁s₂s₃s₂s₁: 3
Also known as: 4-simplex, {3,3,3}. if this polytope has another name.

Special Properties

Universal
Spherical
Locally Spherical
Orientable
Self-Dual

Related Polytopes

Constructions

Facet of

Vertex figure of

Quotients maximal quotients in bold

No regular quotients.

Covers minimal covers in bold

2-fold

4-fold

6-fold

8-fold

10-fold

12-fold

14-fold

16-fold

Irregular Quotients of which this is a minimal cover

None.

Representations

Permutation Representation (GAP)

s0 := (4,5);;
s1 := (3,4);;
s2 := (2,3);;
s3 := (1,2);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP)

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;

Permutation Representation (Magma)

s0 := Sym(5)!(4,5);
s1 := Sym(5)!(3,4);
s2 := Sym(5)!(2,3);
s3 := Sym(5)!(1,2);
poly := sub<Sym(5)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma)

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3 >;

References

Schl�fli, L.; Theorie Der Vielfachen Kontinuit�t, Denkschriften Der Schweizerischen Naturforschenden Gesellschaft, 38, pp1�237 (1901)

to this polytope.