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Polytope of Type {6,9}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,9}*1296c
if this polytope has a name.
Group : SmallGroup(1296,1788)
Rank : 3
Schlafli Type : {6,9}
Number of vertices, edges, etc : 72, 324, 108
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,3}*432
   4-fold quotients : {6,9}*324b
   9-fold quotients : {6,3}*144
   12-fold quotients : {6,3}*108
   27-fold quotients : {6,3}*48
   36-fold quotients : {6,3}*36
   54-fold quotients : {3,3}*24
   108-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,21)(18,22)(19,24)(20,23)
(27,28)(29,33)(30,34)(31,36)(32,35);;
s1 := ( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,29)(14,32)(15,31)(16,30)(17,25)
(18,28)(19,27)(20,26)(21,33)(22,36)(23,35)(24,34);;
s2 := ( 1,14)( 2,13)( 3,15)( 4,16)( 5,22)( 6,21)( 7,23)( 8,24)( 9,18)(10,17)
(11,19)(12,20)(25,26)(29,34)(30,33)(31,35)(32,36);;
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(36)!( 3, 4)( 5, 9)( 6,10)( 7,12)( 8,11)(15,16)(17,21)(18,22)(19,24)
(20,23)(27,28)(29,33)(30,34)(31,36)(32,35);
s1 := Sym(36)!( 2, 4)( 5, 9)( 6,12)( 7,11)( 8,10)(13,29)(14,32)(15,31)(16,30)
(17,25)(18,28)(19,27)(20,26)(21,33)(22,36)(23,35)(24,34);
s2 := Sym(36)!( 1,14)( 2,13)( 3,15)( 4,16)( 5,22)( 6,21)( 7,23)( 8,24)( 9,18)
(10,17)(11,19)(12,20)(25,26)(29,34)(30,33)(31,35)(32,36);
poly := sub<Sym(36)|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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1, 
s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s2*s1 >;

References : None.
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