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

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,6,9}*432
if this polytope has a name.
Group : SmallGroup(432,544)
Rank : 5
Schlafli Type : {2,2,6,9}
Number of vertices, edges, etc : 2, 2, 6, 27, 9
Order of s0s1s2s3s4 : 18
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,2,6,9,2} of size 864
   {2,2,6,9,4} of size 1728
Vertex Figure Of :
   {2,2,2,6,9} of size 864
   {3,2,2,6,9} of size 1296
   {4,2,2,6,9} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,2,2,9}*144, {2,2,6,3}*144
   9-fold quotients : {2,2,2,3}*48
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,2,6,9}*864, {2,4,6,9}*864, {2,2,6,18}*864b
   3-fold covers : {2,2,18,9}*1296, {2,2,6,9}*1296a, {2,2,6,27}*1296, {2,6,6,9}*1296b, {6,2,6,9}*1296
   4-fold covers : {8,2,6,9}*1728, {2,8,6,9}*1728, {4,4,6,9}*1728, {2,2,6,36}*1728b, {4,2,6,18}*1728b, {2,2,12,18}*1728b, {2,4,6,18}*1728b, {2,2,6,9}*1728, {2,2,12,9}*1728
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := ( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,29)(30,31);;
s3 := ( 5, 8)( 6,14)( 7,11)(10,20)(12,15)(13,17)(16,26)(18,21)(19,23)(22,30)
(24,27)(25,28)(29,31);;
s4 := ( 5, 6)( 7,10)( 8,12)( 9,11)(13,16)(14,18)(15,17)(19,22)(20,24)(21,23)
(26,29)(27,28)(30,31);;
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*s1*s0*s1, 
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, 
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(31)!(1,2);
s1 := Sym(31)!(3,4);
s2 := Sym(31)!( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(28,29)(30,31);
s3 := Sym(31)!( 5, 8)( 6,14)( 7,11)(10,20)(12,15)(13,17)(16,26)(18,21)(19,23)
(22,30)(24,27)(25,28)(29,31);
s4 := Sym(31)!( 5, 6)( 7,10)( 8,12)( 9,11)(13,16)(14,18)(15,17)(19,22)(20,24)
(21,23)(26,29)(27,28)(30,31);
poly := sub<Sym(31)|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*s1*s0*s1, 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, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >; 
 

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