Polytope of Type {3,3,3,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,3,3,3}*720
Also Known As : 5-simplex, {3,3,3,3}. if this polytope has another name.
Group : SmallGroup(720,763)
Rank : 5
Schlafli Type : {3,3,3,3}
Number of vertices, edges, etc : 6, 15, 20, 15, 6
Order of s₀s₁s₂s₃s₄ : 6
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 3
Special Properties :
   Universal
   Spherical
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,3,3,3,2} of size 1440
Vertex Figure Of :
   {2,3,3,3,3} of size 1440
Quotients (Maximal Quotients in Boldface) :
   No Regular Quotients.
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,3,3,6}*1440, {3,3,6,3}*1440, {3,6,3,3}*1440, {6,3,3,3}*1440
Permutation Representation (GAP) :

s0 := (4,6);;
s1 := (5,6);;
s2 := (3,5);;
s3 := (2,3);;
s4 := (1,2);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(6)!(4,6);
s1 := Sym(6)!(5,6);
s2 := Sym(6)!(3,5);
s3 := Sym(6)!(2,3);
s4 := Sym(6)!(1,2);
poly := sub<Sym(6)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3, s3*s4*s3*s4*s3*s4 >;

References :

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

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