Polytope of Type {2,21,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,21,6}*504
if this polytope has a name.
Group : SmallGroup(504,192)
Rank : 4
Schlafli Type : {2,21,6}
Number of vertices, edges, etc : 2, 21, 63, 6
Order of s₀s₁s₂s₃ : 42
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,21,6,2} of size 1008
   {2,21,6,3} of size 1512
Vertex Figure Of :
   {2,2,21,6} of size 1008
   {3,2,21,6} of size 1512
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {2,21,2}*168
   7-fold quotients : {2,3,6}*72
   9-fold quotients : {2,7,2}*56
   21-fold quotients : {2,3,2}*24
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,42,6}*1008c
   3-fold covers : {2,63,6}*1512, {2,21,6}*1512, {6,21,6}*1512
Permutation Representation (GAP) :

s0 := (1,2);;
s1 := ( 4, 9)( 5, 8)( 6, 7)(10,17)(11,23)(12,22)(13,21)(14,20)(15,19)(16,18)
(24,45)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,59)(32,65)(33,64)(34,63)
(35,62)(36,61)(37,60)(38,52)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53);;
s2 := ( 3,32)( 4,31)( 5,37)( 6,36)( 7,35)( 8,34)( 9,33)(10,25)(11,24)(12,30)
(13,29)(14,28)(15,27)(16,26)(17,39)(18,38)(19,44)(20,43)(21,42)(22,41)(23,40)
(45,53)(46,52)(47,58)(48,57)(49,56)(50,55)(51,54)(59,60)(61,65)(62,64);;
s3 := (24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)(33,54)
(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)
(44,65);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s3*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*s1*s2*s1*s2 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(65)!(1,2);
s1 := Sym(65)!( 4, 9)( 5, 8)( 6, 7)(10,17)(11,23)(12,22)(13,21)(14,20)(15,19)
(16,18)(24,45)(25,51)(26,50)(27,49)(28,48)(29,47)(30,46)(31,59)(32,65)(33,64)
(34,63)(35,62)(36,61)(37,60)(38,52)(39,58)(40,57)(41,56)(42,55)(43,54)(44,53);
s2 := Sym(65)!( 3,32)( 4,31)( 5,37)( 6,36)( 7,35)( 8,34)( 9,33)(10,25)(11,24)
(12,30)(13,29)(14,28)(15,27)(16,26)(17,39)(18,38)(19,44)(20,43)(21,42)(22,41)
(23,40)(45,53)(46,52)(47,58)(48,57)(49,56)(50,55)(51,54)(59,60)(61,65)(62,64);
s3 := Sym(65)!(24,45)(25,46)(26,47)(27,48)(28,49)(29,50)(30,51)(31,52)(32,53)
(33,54)(34,55)(35,56)(36,57)(37,58)(38,59)(39,60)(40,61)(41,62)(42,63)(43,64)
(44,65);
poly := sub<Sym(65)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, 
s1*s2*s3*s2*s1*s2*s1*s2*s3*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*s1*s2*s1*s2 >;

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