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Polytope of Type {6,21}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,21}*756
if this polytope has a name.
Group : SmallGroup(756,98)
Rank : 3
Schlafli Type : {6,21}
Number of vertices, edges, etc : 18, 189, 63
Order of s0s1s2 : 42
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,21,2} of size 1512
Vertex Figure Of :
   {2,6,21} of size 1512
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,21}*252
   7-fold quotients : {6,3}*108
   9-fold quotients : {2,21}*84
   21-fold quotients : {6,3}*36
   27-fold quotients : {2,7}*28
   63-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,42}*1512b
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)(29,30)
(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)
(62,63);;
s1 := ( 4,19)( 5,20)( 6,21)( 7,16)( 8,17)( 9,18)(10,13)(11,14)(12,15)(22,45)
(23,43)(24,44)(25,63)(26,61)(27,62)(28,60)(29,58)(30,59)(31,57)(32,55)(33,56)
(34,54)(35,52)(36,53)(37,51)(38,49)(39,50)(40,48)(41,46)(42,47);;
s2 := ( 1,25)( 2,26)( 3,27)( 4,22)( 5,23)( 6,24)( 7,40)( 8,41)( 9,42)(10,37)
(11,38)(12,39)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,28)(20,29)(21,30)
(43,46)(44,47)(45,48)(49,61)(50,62)(51,63)(52,58)(53,59)(54,60);;
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*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(63)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(23,24)(26,27)
(29,30)(32,33)(35,36)(38,39)(41,42)(44,45)(47,48)(50,51)(53,54)(56,57)(59,60)
(62,63);
s1 := Sym(63)!( 4,19)( 5,20)( 6,21)( 7,16)( 8,17)( 9,18)(10,13)(11,14)(12,15)
(22,45)(23,43)(24,44)(25,63)(26,61)(27,62)(28,60)(29,58)(30,59)(31,57)(32,55)
(33,56)(34,54)(35,52)(36,53)(37,51)(38,49)(39,50)(40,48)(41,46)(42,47);
s2 := Sym(63)!( 1,25)( 2,26)( 3,27)( 4,22)( 5,23)( 6,24)( 7,40)( 8,41)( 9,42)
(10,37)(11,38)(12,39)(13,34)(14,35)(15,36)(16,31)(17,32)(18,33)(19,28)(20,29)
(21,30)(43,46)(44,47)(45,48)(49,61)(50,62)(51,63)(52,58)(53,59)(54,60);
poly := sub<Sym(63)|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*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
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