Polytope of Type {28,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {28,4}*1008
if this polytope has a name.
Group : SmallGroup(1008,896)
Rank : 3
Schlafli Type : {28,4}
Number of vertices, edges, etc : 126, 252, 18
Order of s0s1s2 : 42
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   7-fold quotients : {4,4}*144
   14-fold quotients : {4,4}*72
   18-fold quotients : {14,2}*56
   36-fold quotients : {7,2}*28
   126-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 2.
      9 facets:
         9 of {28}*56
      63 vertex figures:
         63 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      9 facets:
         9 of {28}*56
      63 vertex figures:
         63 of {4}*8
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      10 facets:
         2 of {14}*28
         8 of {28}*56
      70 vertex figures:
         56 of {4}*8
         14 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2> of order 3.
      6 facets:
         6 of {28}*56
      42 vertex figures:
         42 of {4}*8
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
      5 facets:
         1 of {14}*28
         4 of {28}*56
      35 vertex figures:
         28 of {4}*8
         7 of {2}*4
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 6.
      4 facets:
         2 of {14}*28
         2 of {28}*56
      28 vertex figures:
         14 of {4}*8
         14 of {2}*4

Permutation Representation (GAP) : 
s0 := ( 2, 7)( 3, 6)( 4, 5)( 8,22)( 9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,43)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(30,35)(31,34)(32,33)(36,50)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(58,63)(59,62)(60,61);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 8, 9)(10,14)(11,13)(15,16)(17,21)(18,20)(22,44)(23,43)(24,49)(25,48)(26,47)(27,46)(28,45)(29,51)(30,50)(31,56)(32,55)(33,54)(34,53)(35,52)(36,58)(37,57)(38,63)(39,62)(40,61)(41,60)(42,59);;
s2 := ( 1,29)( 2,30)( 3,31)( 4,32)( 5,33)( 6,34)( 7,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49);;
poly := Group([s0,s1,s2]);;
Finitely Presented Group Representation (GAP) : 
F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(63)!( 2, 7)( 3, 6)( 4, 5)( 8,22)( 9,28)(10,27)(11,26)(12,25)(13,24)(14,23)(15,43)(16,49)(17,48)(18,47)(19,46)(20,45)(21,44)(30,35)(31,34)(32,33)(36,50)(37,56)(38,55)(39,54)(40,53)(41,52)(42,51)(58,63)(59,62)(60,61);
s1 := Sym(63)!( 1, 2)( 3, 7)( 4, 6)( 8, 9)(10,14)(11,13)(15,16)(17,21)(18,20)(22,44)(23,43)(24,49)(25,48)(26,47)(27,46)(28,45)(29,51)(30,50)(31,56)(32,55)(33,54)(34,53)(35,52)(36,58)(37,57)(38,63)(39,62)(40,61)(41,60)(42,59);
s2 := Sym(63)!( 1,29)( 2,30)( 3,31)( 4,32)( 5,33)( 6,34)( 7,35)(15,50)(16,51)(17,52)(18,53)(19,54)(20,55)(21,56)(36,43)(37,44)(38,45)(39,46)(40,47)(41,48)(42,49);
poly := sub<Sym(63)|s0,s1,s2>;
Finitely Presented Group Representation (Magma) : 
poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2 >;
References : None.
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