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Polytope of Type {3,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6}*432
Also Known As : {3,6}(6,0), {3,6}12if this polytope has another name.
Group : SmallGroup(432,523)
Rank : 3
Schlafli Type : {3,6}
Number of vertices, edges, etc : 36, 108, 72
Order of s0s1s2 : 12
Order of s0s1s2s1 : 6
Special Properties :
   Toroidal
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {3,6,2} of size 864
   {3,6,4} of size 1728
Vertex Figure Of :
   {2,3,6} of size 864
   {4,3,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,6}*144
   4-fold quotients : {3,6}*108
   9-fold quotients : {3,6}*48
   12-fold quotients : {3,6}*36
   18-fold quotients : {3,3}*24
   36-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,12}*864, {6,6}*864a
   3-fold covers : {9,6}*1296a, {3,6}*1296, {9,6}*1296b, {3,18}*1296a, {9,6}*1296c, {9,6}*1296d
   4-fold covers : {3,6}*1728, {12,6}*1728a, {6,12}*1728c, {6,6}*1728b, {12,6}*1728d, {6,12}*1728e, {3,12}*1728
Permutation Representation (GAP) :
s0 := ( 2, 3)( 6, 7)(10,11)(13,25)(14,27)(15,26)(16,28)(17,29)(18,31)(19,30)
(20,32)(21,33)(22,35)(23,34)(24,36);;
s1 := ( 1,17)( 2,20)( 3,19)( 4,18)( 5,21)( 6,24)( 7,23)( 8,22)( 9,13)(10,16)
(11,15)(12,14)(26,28)(30,32)(34,36);;
s2 := ( 1, 4)( 5,12)( 6,10)( 7,11)( 8, 9)(13,16)(17,24)(18,22)(19,23)(20,21)
(25,28)(29,36)(30,34)(31,35)(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, 
s1*s2*s1*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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(36)!( 2, 3)( 6, 7)(10,11)(13,25)(14,27)(15,26)(16,28)(17,29)(18,31)
(19,30)(20,32)(21,33)(22,35)(23,34)(24,36);
s1 := Sym(36)!( 1,17)( 2,20)( 3,19)( 4,18)( 5,21)( 6,24)( 7,23)( 8,22)( 9,13)
(10,16)(11,15)(12,14)(26,28)(30,32)(34,36);
s2 := Sym(36)!( 1, 4)( 5,12)( 6,10)( 7,11)( 8, 9)(13,16)(17,24)(18,22)(19,23)
(20,21)(25,28)(29,36)(30,34)(31,35)(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, s1*s2*s1*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*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >; 
 
References : None.
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