Polytope of Type {3,4,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,4,3}*576
Also Known As : hemi-24-cell, {3,4,3}₆. if this polytope has another name.
Group : SmallGroup(576,8654)
Rank : 4
Schlafli Type : {3,4,3}
Number of vertices, edges, etc : 12, 48, 48, 12
Order of s₀s₁s₂s₃ : 6
Order of s₀s₁s₂s₃s₂s₁ : 4
Special Properties :
   Projective
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,4,3,2} of size 1152
Vertex Figure Of :
   {2,3,4,3} of size 1152
Quotients (Maximal Quotients in Boldface) :
   16-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,4,3}*1152, {3,4,6}*1152a, {6,4,3}*1152a, {3,4,6}*1152b, {6,4,3}*1152b
   3-fold covers : {3,4,9}*1728, {9,4,3}*1728, {3,12,3}*1728
Permutation Representation (GAP) :

s0 := ( 2, 3)( 7, 8)(10,12);;
s1 := ( 2, 4)( 6, 7)(11,12);;
s2 := ( 5,11)( 6, 9)( 7,10)( 8,12);;
s3 := ( 1, 9)( 2,12)( 3,10)( 4,11);;
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, 
s2*s3*s2*s3*s2*s3, s1*s2*s1*s2*s1*s2*s1*s2, 
s3*s0*s2*s1*s0*s2*s1*s3*s2*s3*s1*s2*s0*s1*s3*s2*s3*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(12)!( 2, 3)( 7, 8)(10,12);
s1 := Sym(12)!( 2, 4)( 6, 7)(11,12);
s2 := Sym(12)!( 5,11)( 6, 9)( 7,10)( 8,12);
s3 := Sym(12)!( 1, 9)( 2,12)( 3,10)( 4,11);
poly := sub<Sym(12)|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, s2*s3*s2*s3*s2*s3, 
s1*s2*s1*s2*s1*s2*s1*s2, s3*s0*s2*s1*s0*s2*s1*s3*s2*s3*s1*s2*s0*s1*s3*s2*s3*s1*s2*s0*s1*s2*s0*s1 >;

References : None.
to this polytope