Polytope of Type {12,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,4}*1728e
if this polytope has a name.
Group : SmallGroup(1728,47847)
Rank : 3
Schlafli Type : {12,4}
Number of vertices, edges, etc : 216, 432, 72
Order of s₀s₁s₂ : 6
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 : {12,4}*432b
   9-fold quotients : {12,4}*192c
   12-fold quotients : {4,4}*144
   18-fold quotients : {6,4}*96
   24-fold quotients : {4,4}*72
   36-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   72-fold quotients : {3,4}*24, {6,2}*24
   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)( 5,29)( 6,30)( 7,32)( 8,31)( 9,21)(10,22)(11,24)(12,23)(13,25)
(14,26)(15,28)(16,27)(19,20)(35,36);;
s1 := ( 2, 4)( 6, 8)(10,12)(13,33)(14,36)(15,35)(16,34)(17,25)(18,28)(19,27)
(20,26)(21,29)(22,32)(23,31)(24,30);;
s2 := ( 1,18)( 2,17)( 3,20)( 4,19)( 5,26)( 6,25)( 7,28)( 8,27)( 9,10)(11,12)
(13,30)(14,29)(15,32)(16,31)(21,22)(23,24)(33,34)(35,36);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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