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Polytope of Type {4,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,12}*1728e
if this polytope has a name.
Group : SmallGroup(1728,47847)
Rank : 3
Schlafli Type : {4,12}
Number of vertices, edges, etc : 72, 432, 216
Order of s0s1s2 : 6
Order of s0s1s2s1 : 12
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   4-fold quotients : {4,12}*432b
   9-fold quotients : {4,12}*192c
   12-fold quotients : {4,4}*144
   18-fold quotients : {4,6}*96
   24-fold quotients : {4,4}*72
   36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   72-fold quotients : {4,3}*24, {2,6}*24
   144-fold quotients : {2,3}*12
   216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 3)( 2, 4)( 5,31)( 6,32)( 7,29)( 8,30)( 9,23)(10,24)(11,21)(12,22)
(13,27)(14,28)(15,25)(16,26)(17,19)(18,20)(33,35)(34,36);;
s1 := ( 3, 4)( 7, 8)(11,12)(13,33)(14,34)(15,36)(16,35)(17,25)(18,26)(19,28)
(20,27)(21,29)(22,30)(23,32)(24,31);;
s2 := ( 1,17)( 2,20)( 3,19)( 4,18)( 5,25)( 6,28)( 7,27)( 8,26)(10,12)(13,29)
(14,32)(15,31)(16,30)(22,24)(34,36);;
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(36)!( 1, 3)( 2, 4)( 5,31)( 6,32)( 7,29)( 8,30)( 9,23)(10,24)(11,21)
(12,22)(13,27)(14,28)(15,25)(16,26)(17,19)(18,20)(33,35)(34,36);
s1 := Sym(36)!( 3, 4)( 7, 8)(11,12)(13,33)(14,34)(15,36)(16,35)(17,25)(18,26)
(19,28)(20,27)(21,29)(22,30)(23,32)(24,31);
s2 := Sym(36)!( 1,17)( 2,20)( 3,19)( 4,18)( 5,25)( 6,28)( 7,27)( 8,26)(10,12)
(13,29)(14,32)(15,31)(16,30)(22,24)(34,36);
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, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s0*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1 >; 
 
References : None.
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