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Polytope of Type {3,2,22,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,2,22,6}*1584
if this polytope has a name.
Group : SmallGroup(1584,675)
Rank : 5
Schlafli Type : {3,2,22,6}
Number of vertices, edges, etc : 3, 3, 22, 66, 6
Order of s0s1s2s3s4 : 66
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,22,2}*528
   6-fold quotients : {3,2,11,2}*264
   11-fold quotients : {3,2,2,6}*144
   22-fold quotients : {3,2,2,3}*72
   33-fold quotients : {3,2,2,2}*48
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (2,3);;
s1 := (1,2);;
s2 := ( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(16,25)(17,24)(18,23)(19,22)(20,21)
(27,36)(28,35)(29,34)(30,33)(31,32)(38,47)(39,46)(40,45)(41,44)(42,43)(49,58)
(50,57)(51,56)(52,55)(53,54)(60,69)(61,68)(62,67)(63,66)(64,65);;
s3 := ( 4, 5)( 6,14)( 7,13)( 8,12)( 9,11)(15,27)(16,26)(17,36)(18,35)(19,34)
(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(37,38)(39,47)(40,46)(41,45)(42,44)
(48,60)(49,59)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)
(58,61);;
s4 := ( 4,48)( 5,49)( 6,50)( 7,51)( 8,52)( 9,53)(10,54)(11,55)(12,56)(13,57)
(14,58)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)(24,46)
(25,47)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)(35,68)
(36,69);;
poly := Group([s0,s1,s2,s3,s4]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s1*s0*s1*s0*s1, s2*s3*s4*s3*s2*s3*s4*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(69)!(2,3);
s1 := Sym(69)!(1,2);
s2 := Sym(69)!( 5,14)( 6,13)( 7,12)( 8,11)( 9,10)(16,25)(17,24)(18,23)(19,22)
(20,21)(27,36)(28,35)(29,34)(30,33)(31,32)(38,47)(39,46)(40,45)(41,44)(42,43)
(49,58)(50,57)(51,56)(52,55)(53,54)(60,69)(61,68)(62,67)(63,66)(64,65);
s3 := Sym(69)!( 4, 5)( 6,14)( 7,13)( 8,12)( 9,11)(15,27)(16,26)(17,36)(18,35)
(19,34)(20,33)(21,32)(22,31)(23,30)(24,29)(25,28)(37,38)(39,47)(40,46)(41,45)
(42,44)(48,60)(49,59)(50,69)(51,68)(52,67)(53,66)(54,65)(55,64)(56,63)(57,62)
(58,61);
s4 := Sym(69)!( 4,48)( 5,49)( 6,50)( 7,51)( 8,52)( 9,53)(10,54)(11,55)(12,56)
(13,57)(14,58)(15,37)(16,38)(17,39)(18,40)(19,41)(20,42)(21,43)(22,44)(23,45)
(24,46)(25,47)(26,59)(27,60)(28,61)(29,62)(30,63)(31,64)(32,65)(33,66)(34,67)
(35,68)(36,69);
poly := sub<Sym(69)|s0,s1,s2,s3,s4>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s0*s1*s0*s1, 
s2*s3*s4*s3*s2*s3*s4*s3, s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >; 
 

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