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Polytope of Type {8,2,6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,2,6,9}*1728
if this polytope has a name.
Group : SmallGroup(1728,15957)
Rank : 5
Schlafli Type : {8,2,6,9}
Number of vertices, edges, etc : 8, 8, 6, 27, 9
Order of s0s1s2s3s4 : 72
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,2,6,9}*864
   3-fold quotients : {8,2,2,9}*576, {8,2,6,3}*576
   4-fold quotients : {2,2,6,9}*432
   6-fold quotients : {4,2,2,9}*288, {4,2,6,3}*288
   9-fold quotients : {8,2,2,3}*192
   12-fold quotients : {2,2,2,9}*144, {2,2,6,3}*144
   18-fold quotients : {4,2,2,3}*96
   36-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3)(4,5)(6,7);;
s1 := (1,2)(3,4)(5,6)(7,8);;
s2 := (12,13)(15,16)(18,19)(21,22)(24,25)(27,28)(30,31)(32,33)(34,35);;
s3 := ( 9,12)(10,18)(11,15)(14,24)(16,19)(17,21)(20,30)(22,25)(23,27)(26,34)
(28,31)(29,32)(33,35);;
s4 := ( 9,10)(11,14)(12,16)(13,15)(17,20)(18,22)(19,21)(23,26)(24,28)(25,27)
(30,33)(31,32)(34,35);;
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, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(35)!(2,3)(4,5)(6,7);
s1 := Sym(35)!(1,2)(3,4)(5,6)(7,8);
s2 := Sym(35)!(12,13)(15,16)(18,19)(21,22)(24,25)(27,28)(30,31)(32,33)(34,35);
s3 := Sym(35)!( 9,12)(10,18)(11,15)(14,24)(16,19)(17,21)(20,30)(22,25)(23,27)
(26,34)(28,31)(29,32)(33,35);
s4 := Sym(35)!( 9,10)(11,14)(12,16)(13,15)(17,20)(18,22)(19,21)(23,26)(24,28)
(25,27)(30,33)(31,32)(34,35);
poly := sub<Sym(35)|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, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s4*s2*s3*s2*s3*s4*s2*s3*s2*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >; 
 

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