Polytope of Type {3,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,3}*720a
Also Known As : 7T4(2,0)(2,2). if this polytope has another name.
Group : SmallGroup(720,767)
Rank : 4
Schlafli Type : {3,6,3}
Number of vertices, edges, etc : 5, 60, 60, 15
Order of s0s1s2s3 : 15
Order of s0s1s2s3s2s1 : 6
Special Properties :
   Universal
   Locally Toroidal
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,3,2} of size 1440
Vertex Figure Of :
   {2,3,6,3} of size 1440
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6,3}*240
   6-fold quotients : {3,3,3}*120
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,12,3}*1440a, {3,6,6}*1440a, {6,6,3}*1440b
Permutation Representation (GAP) : 
s0 := (7,8);;
s1 := (6,7);;
s2 := (2,3)(5,6);;
s3 := (1,2)(4,5);;
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, s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(8)!(7,8);
s1 := Sym(8)!(6,7);
s2 := Sym(8)!(2,3)(5,6);
s3 := Sym(8)!(1,2)(4,5);
poly := sub<Sym(8)|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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References :
  1. Theorem 11E5,11E10, McMullen P., Schulte, E.; Abstract Regular Polytopes \ (Cambridge University Press, 2002)

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