Polytope of Type {3,6,21}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,21}*756
if this polytope has a name.
Group : SmallGroup(756,98)
Rank : 4
Schlafli Type : {3,6,21}
Number of vertices, edges, etc : 3, 9, 63, 21
Order of s₀s₁s₂s₃ : 21
Order of s₀s₁s₂s₃s₂s₁ : 6
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {3,6,21,2} of size 1512
Vertex Figure Of :
   {2,3,6,21} of size 1512
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {3,2,21}*252
   7-fold quotients : {3,6,3}*108
   9-fold quotients : {3,2,7}*84
   21-fold quotients : {3,2,3}*36
Covers (Minimal Covers in Boldface) :
   2-fold covers : {3,6,42}*1512a, {6,6,21}*1512a
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 := ( 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);;
s2 := ( 1,22)( 2,24)( 3,23)( 4,40)( 5,42)( 6,41)( 7,37)( 8,39)( 9,38)(10,34)
(11,36)(12,35)(13,31)(14,33)(15,32)(16,28)(17,30)(18,29)(19,25)(20,27)(21,26)
(44,45)(46,61)(47,63)(48,62)(49,58)(50,60)(51,59)(52,55)(53,57)(54,56);;
s3 := ( 1, 4)( 2, 6)( 3, 5)( 7,19)( 8,21)( 9,20)(10,16)(11,18)(12,17)(14,15)
(22,46)(23,48)(24,47)(25,43)(26,45)(27,44)(28,61)(29,63)(30,62)(31,58)(32,60)
(33,59)(34,55)(35,57)(36,56)(37,52)(38,54)(39,53)(40,49)(41,51)(42,50);;
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, s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 
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(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)!( 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);
s2 := Sym(63)!( 1,22)( 2,24)( 3,23)( 4,40)( 5,42)( 6,41)( 7,37)( 8,39)( 9,38)
(10,34)(11,36)(12,35)(13,31)(14,33)(15,32)(16,28)(17,30)(18,29)(19,25)(20,27)
(21,26)(44,45)(46,61)(47,63)(48,62)(49,58)(50,60)(51,59)(52,55)(53,57)(54,56);
s3 := Sym(63)!( 1, 4)( 2, 6)( 3, 5)( 7,19)( 8,21)( 9,20)(10,16)(11,18)(12,17)
(14,15)(22,46)(23,48)(24,47)(25,43)(26,45)(27,44)(28,61)(29,63)(30,62)(31,58)
(32,60)(33,59)(34,55)(35,57)(36,56)(37,52)(38,54)(39,53)(40,49)(41,51)(42,50);
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, 
s0*s1*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s1*s2*s1*s2*s3*s1*s2*s1*s2, 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 >;

References : None.
to this polytope