Polytope of Type {6,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,6}*864c
if this polytope has a name.
Group : SmallGroup(864,4673)
Rank : 3
Schlafli Type : {6,6}
Number of vertices, edges, etc : 72, 216, 72
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {6,6,2} of size 1728
Vertex Figure Of :
   {2,6,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,6}*288a, {6,6}*288b
   4-fold quotients : {6,6}*216d
   6-fold quotients : {3,6}*144, {6,3}*144
   9-fold quotients : {6,6}*96
   12-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   18-fold quotients : {3,6}*48, {6,3}*48
   24-fold quotients : {3,6}*36, {6,3}*36
   36-fold quotients : {3,3}*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 : {6,12}*1728g, {12,6}*1728g, {6,6}*1728f, {6,12}*1728h, {12,6}*1728h
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)( 2,14)( 3,15)( 4,13)( 5,20)( 6,18)( 7,19)( 8,17)( 9,24)(10,22)
(11,23)(12,21)(25,28)(29,32)(33,36);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*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)( 2,14)( 3,15)( 4,13)( 5,20)( 6,18)( 7,19)( 8,17)( 9,24)
(10,22)(11,23)(12,21)(25,28)(29,32)(33,36);
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1 >;

References : None.
to this polytope