Polytope of Type {6,3,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,3,3}*1296
Also Known As : 3T4(3,0), {{6,3}6,{3,3}}. if this polytope has another name.
Group : SmallGroup(1296,3490)
Rank : 4
Schlafli Type : {6,3,3}
Number of vertices, edges, etc : 54, 108, 54, 12
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 6
Special Properties :
   Universal
   Locally Toroidal
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   27-fold quotients : {2,3,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12);;
s1 := (1,9)(2,8)(3,7);;
s2 := ( 7,10)( 8,11)( 9,12);;
s3 := ( 4,10)( 5,12)( 6,11);;
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, s1*s2*s1*s2*s1*s2, 
s2*s3*s2*s3*s2*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(12)!( 2, 3)( 5, 6)( 8, 9)(11,12);
s1 := Sym(12)!(1,9)(2,8)(3,7);
s2 := Sym(12)!( 7,10)( 8,11)( 9,12);
s3 := Sym(12)!( 4,10)( 5,12)( 6,11);
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, 
s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1 >;
References :
  1. Theorem 11B5, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambr\ idge University Press, 2002)

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