Polytope of Type {21,6,3}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,6,3}*756
if this polytope has a name.
Group : SmallGroup(756,98)
Rank : 4
Schlafli Type : {21,6,3}
Number of vertices, edges, etc : 21, 63, 9, 3
Order of s₀s₁s₂s₃ : 21
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {21,6,3,2} of size 1512
Vertex Figure Of :
   {2,21,6,3} of size 1512
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {21,2,3}*252
   7-fold quotients : {3,6,3}*108
   9-fold quotients : {7,2,3}*84
   21-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {21,6,6}*1512a, {42,6,3}*1512a
Permutation Representation (GAP) :

s0 := ( 2, 3)( 4,19)( 5,21)( 6,20)( 7,16)( 8,18)( 9,17)(10,13)(11,15)(12,14)
(22,43)(23,45)(24,44)(25,61)(26,63)(27,62)(28,58)(29,60)(30,59)(31,55)(32,57)
(33,56)(34,52)(35,54)(36,53)(37,49)(38,51)(39,50)(40,46)(41,48)(42,47);;
s1 := ( 1,25)( 2,27)( 3,26)( 4,22)( 5,24)( 6,23)( 7,40)( 8,42)( 9,41)(10,37)
(11,39)(12,38)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,28)(20,30)(21,29)
(43,46)(44,48)(45,47)(49,61)(50,63)(51,62)(52,58)(53,60)(54,59)(56,57);;
s2 := ( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(22,23)(25,26)(28,29)
(31,32)(34,35)(37,38)(40,41)(43,45)(46,48)(49,51)(52,54)(55,57)(58,60)
(61,63);;
s3 := ( 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);;
poly := Group([s0,s1,s2,s3]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s2*s3*s2*s3*s2*s3, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(63)!( 2, 3)( 4,19)( 5,21)( 6,20)( 7,16)( 8,18)( 9,17)(10,13)(11,15)
(12,14)(22,43)(23,45)(24,44)(25,61)(26,63)(27,62)(28,58)(29,60)(30,59)(31,55)
(32,57)(33,56)(34,52)(35,54)(36,53)(37,49)(38,51)(39,50)(40,46)(41,48)(42,47);
s1 := Sym(63)!( 1,25)( 2,27)( 3,26)( 4,22)( 5,24)( 6,23)( 7,40)( 8,42)( 9,41)
(10,37)(11,39)(12,38)(13,34)(14,36)(15,35)(16,31)(17,33)(18,32)(19,28)(20,30)
(21,29)(43,46)(44,48)(45,47)(49,61)(50,63)(51,62)(52,58)(53,60)(54,59)(56,57);
s2 := Sym(63)!( 2, 3)( 5, 6)( 8, 9)(11,12)(14,15)(17,18)(20,21)(22,23)(25,26)
(28,29)(31,32)(34,35)(37,38)(40,41)(43,45)(46,48)(49,51)(52,54)(55,57)(58,60)
(61,63);
s3 := 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);
poly := sub<Sym(63)|s0,s1,s2,s3>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2, 
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope