Polytope of Type {6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,12}*864c
if this polytope has a name.
Group : SmallGroup(864,4673)
Rank : 3
Schlafli Type : {6,12}
Number of vertices, edges, etc : 36, 216, 72
Order of s₀s₁s₂ : 6
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,12,2} of size 1728
Vertex Figure Of :
   {2,6,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,12}*288a, {6,12}*288b
   4-fold quotients : {6,6}*216d
   6-fold quotients : {3,12}*144, {6,12}*144d
   9-fold quotients : {6,4}*96
   12-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   18-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   24-fold quotients : {3,6}*36, {6,3}*36
   36-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
   72-fold quotients : {2,3}*12, {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,24}*1728f, {6,24}*1728g, {12,12}*1728v, {6,12}*1728i, {12,12}*1728x
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
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