Polytope of Type {6,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,4}*1728
if this polytope has a name.
Group : SmallGroup(1728,47847)
Rank : 3
Schlafli Type : {6,4}
Number of vertices, edges, etc : 216, 432, 144
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {6,4}*432b
   9-fold quotients : {6,4}*192b
   12-fold quotients : {6,4}*144
   18-fold quotients : {6,4}*96
   24-fold quotients : {6,4}*72
   36-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
   72-fold quotients : {3,4}*24, {6,2}*24
   108-fold quotients : {2,4}*16
   144-fold quotients : {3,2}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Permutation Representation (Magma) :

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

References : None.
to this polytope