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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,2,6,2,6}*1728
if this polytope has a name.
Group : SmallGroup(1728,47915)
Rank : 6
Schlafli Type : {6,2,6,2,6}
Number of vertices, edges, etc : 6, 6, 6, 6, 6, 6
Order of s0s1s2s3s4s5 : 6
Order of s0s1s2s3s4s5s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
   Self-Dual
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 : {3,2,6,2,6}*864, {6,2,3,2,6}*864, {6,2,6,2,3}*864
   3-fold quotients : {2,2,6,2,6}*576, {6,2,2,2,6}*576, {6,2,6,2,2}*576
   4-fold quotients : {3,2,3,2,6}*432, {3,2,6,2,3}*432, {6,2,3,2,3}*432
   6-fold quotients : {2,2,3,2,6}*288, {2,2,6,2,3}*288, {3,2,2,2,6}*288, {3,2,6,2,2}*288, {6,2,2,2,3}*288, {6,2,3,2,2}*288
   8-fold quotients : {3,2,3,2,3}*216
   9-fold quotients : {2,2,2,2,6}*192, {2,2,6,2,2}*192, {6,2,2,2,2}*192
   12-fold quotients : {2,2,3,2,3}*144, {3,2,2,2,3}*144, {3,2,3,2,2}*144
   18-fold quotients : {2,2,2,2,3}*96, {2,2,3,2,2}*96, {3,2,2,2,2}*96
   27-fold quotients : {2,2,2,2,2}*64
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (3,4)(5,6);;
s1 := (1,5)(2,3)(4,6);;
s2 := ( 9,10)(11,12);;
s3 := ( 7,11)( 8, 9)(10,12);;
s4 := (15,16)(17,18);;
s5 := (13,17)(14,15)(16,18);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
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, s3*s4*s3*s4, s0*s5*s0*s5, 
s1*s5*s1*s5, s2*s5*s2*s5, s3*s5*s3*s5, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(18)!(3,4)(5,6);
s1 := Sym(18)!(1,5)(2,3)(4,6);
s2 := Sym(18)!( 9,10)(11,12);
s3 := Sym(18)!( 7,11)( 8, 9)(10,12);
s4 := Sym(18)!(15,16)(17,18);
s5 := Sym(18)!(13,17)(14,15)(16,18);
poly := sub<Sym(18)|s0,s1,s2,s3,s4,s5>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, 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, s3*s4*s3*s4, 
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5, 
s3*s5*s3*s5, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >; 
 

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