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Polytope of Type {6,9,2,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9,2,3}*648
if this polytope has a name.
Group : SmallGroup(648,554)
Rank : 5
Schlafli Type : {6,9,2,3}
Number of vertices, edges, etc : 6, 27, 9, 3, 3
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,9,2,3,2} of size 1296
Vertex Figure Of :
   {2,6,9,2,3} of size 1296
   {3,6,9,2,3} of size 1944
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,9,2,3}*216, {6,3,2,3}*216
   9-fold quotients : {2,3,2,3}*72
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,9,2,6}*1296, {6,18,2,3}*1296b
   3-fold covers : {6,9,2,9}*1944, {18,9,2,3}*1944, {6,9,2,3}*1944a, {6,9,6,3}*1944, {6,27,2,3}*1944
Permutation Representation (GAP) :
s0 := ( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(26,27);;
s1 := ( 1, 4)( 2,10)( 3, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)(18,26)
(20,23)(21,24)(25,27);;
s2 := ( 1, 2)( 3, 6)( 4, 8)( 5, 7)( 9,12)(10,14)(11,13)(15,18)(16,20)(17,19)
(22,25)(23,24)(26,27);;
s3 := (29,30);;
s4 := (28,29);;
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, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s3*s4*s3*s4*s3*s4, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(30)!( 4, 5)( 7, 8)(10,11)(13,14)(16,17)(19,20)(22,23)(24,25)(26,27);
s1 := Sym(30)!( 1, 4)( 2,10)( 3, 7)( 6,16)( 8,11)( 9,13)(12,22)(14,17)(15,19)
(18,26)(20,23)(21,24)(25,27);
s2 := Sym(30)!( 1, 2)( 3, 6)( 4, 8)( 5, 7)( 9,12)(10,14)(11,13)(15,18)(16,20)
(17,19)(22,25)(23,24)(26,27);
s3 := Sym(30)!(29,30);
s4 := Sym(30)!(28,29);
poly := sub<Sym(30)|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, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s3*s4*s3*s4*s3*s4, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 

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