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Polytope of Type {2,2,2,6,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,2,2,6,18}*1728b
if this polytope has a name.
Group : SmallGroup(1728,46164)
Rank : 6
Schlafli Type : {2,2,2,6,18}
Number of vertices, edges, etc : 2, 2, 2, 6, 54, 18
Order of s0s1s2s3s4s5 : 18
Order of s0s1s2s3s4s5s4s3s2s1 : 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) :
   2-fold quotients : {2,2,2,6,9}*864
   3-fold quotients : {2,2,2,2,18}*576, {2,2,2,6,6}*576b
   6-fold quotients : {2,2,2,2,9}*288, {2,2,2,6,3}*288
   9-fold quotients : {2,2,2,2,6}*192
   18-fold quotients : {2,2,2,2,3}*96
   27-fold quotients : {2,2,2,2,2}*64
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (3,4);;
s2 := (5,6);;
s3 := (10,13)(11,14)(12,15)(19,22)(20,23)(21,24)(28,31)(29,32)(30,33)(37,40)
(38,41)(39,42)(46,49)(47,50)(48,51)(55,58)(56,59)(57,60);;
s4 := ( 7,10)( 8,12)( 9,11)(14,15)(16,29)(17,28)(18,30)(19,26)(20,25)(21,27)
(22,32)(23,31)(24,33)(34,37)(35,39)(36,38)(41,42)(43,56)(44,55)(45,57)(46,53)
(47,52)(48,54)(49,59)(50,58)(51,60);;
s5 := ( 7,43)( 8,45)( 9,44)(10,49)(11,51)(12,50)(13,46)(14,48)(15,47)(16,34)
(17,36)(18,35)(19,40)(20,42)(21,41)(22,37)(23,39)(24,38)(25,53)(26,52)(27,54)
(28,59)(29,58)(30,60)(31,56)(32,55)(33,57);;
poly := Group([s0,s1,s2,s3,s4,s5]);;
 
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4","s5");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  s5 := F.6;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s5*s5, 
s0*s1*s0*s1, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s0*s5*s0*s5, s1*s5*s1*s5, s2*s5*s2*s5, 
s3*s5*s3*s5, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(60)!(1,2);
s1 := Sym(60)!(3,4);
s2 := Sym(60)!(5,6);
s3 := Sym(60)!(10,13)(11,14)(12,15)(19,22)(20,23)(21,24)(28,31)(29,32)(30,33)
(37,40)(38,41)(39,42)(46,49)(47,50)(48,51)(55,58)(56,59)(57,60);
s4 := Sym(60)!( 7,10)( 8,12)( 9,11)(14,15)(16,29)(17,28)(18,30)(19,26)(20,25)
(21,27)(22,32)(23,31)(24,33)(34,37)(35,39)(36,38)(41,42)(43,56)(44,55)(45,57)
(46,53)(47,52)(48,54)(49,59)(50,58)(51,60);
s5 := Sym(60)!( 7,43)( 8,45)( 9,44)(10,49)(11,51)(12,50)(13,46)(14,48)(15,47)
(16,34)(17,36)(18,35)(19,40)(20,42)(21,41)(22,37)(23,39)(24,38)(25,53)(26,52)
(27,54)(28,59)(29,58)(30,60)(31,56)(32,55)(33,57);
poly := sub<Sym(60)|s0,s1,s2,s3,s4,s5>;
 
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4,s5> := Group< s0,s1,s2,s3,s4,s5 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s5*s5, s0*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3, s0*s4*s0*s4, s1*s4*s1*s4, 
s2*s4*s2*s4, s0*s5*s0*s5, s1*s5*s1*s5, 
s2*s5*s2*s5, s3*s5*s3*s5, s5*s3*s4*s3*s4*s5*s3*s4*s3*s4, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4, 
s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5*s4*s5 >; 
 

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