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Polytope of Type {21,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {21,6}*1008b
if this polytope has a name.
Group : SmallGroup(1008,904)
Rank : 3
Schlafli Type : {21,6}
Number of vertices, edges, etc : 84, 252, 24
Order of s0s1s2 : 84
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {21,6}*336
   4-fold quotients : {21,6}*252
   7-fold quotients : {3,6}*144
   12-fold quotients : {21,2}*84
   21-fold quotients : {3,6}*48
   28-fold quotients : {3,6}*36
   36-fold quotients : {7,2}*28
   42-fold quotients : {3,3}*24
   84-fold quotients : {3,2}*12
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5,25)( 6,26)( 7,28)( 8,27)( 9,21)(10,22)(11,24)(12,23)(13,17)
(14,18)(15,20)(16,19)(29,57)(30,58)(31,60)(32,59)(33,81)(34,82)(35,84)(36,83)
(37,77)(38,78)(39,80)(40,79)(41,73)(42,74)(43,76)(44,75)(45,69)(46,70)(47,72)
(48,71)(49,65)(50,66)(51,68)(52,67)(53,61)(54,62)(55,64)(56,63);;
s1 := ( 1,33)( 2,36)( 3,35)( 4,34)( 5,29)( 6,32)( 7,31)( 8,30)( 9,53)(10,56)
(11,55)(12,54)(13,49)(14,52)(15,51)(16,50)(17,45)(18,48)(19,47)(20,46)(21,41)
(22,44)(23,43)(24,42)(25,37)(26,40)(27,39)(28,38)(57,61)(58,64)(59,63)(60,62)
(65,81)(66,84)(67,83)(68,82)(69,77)(70,80)(71,79)(72,78)(74,76);;
s2 := ( 1, 2)( 5, 6)( 9,10)(13,14)(17,18)(21,22)(25,26)(29,30)(33,34)(37,38)
(41,42)(45,46)(49,50)(53,54)(57,58)(61,62)(65,66)(69,70)(73,74)(77,78)
(81,82);;
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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*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*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(84)!( 3, 4)( 5,25)( 6,26)( 7,28)( 8,27)( 9,21)(10,22)(11,24)(12,23)
(13,17)(14,18)(15,20)(16,19)(29,57)(30,58)(31,60)(32,59)(33,81)(34,82)(35,84)
(36,83)(37,77)(38,78)(39,80)(40,79)(41,73)(42,74)(43,76)(44,75)(45,69)(46,70)
(47,72)(48,71)(49,65)(50,66)(51,68)(52,67)(53,61)(54,62)(55,64)(56,63);
s1 := Sym(84)!( 1,33)( 2,36)( 3,35)( 4,34)( 5,29)( 6,32)( 7,31)( 8,30)( 9,53)
(10,56)(11,55)(12,54)(13,49)(14,52)(15,51)(16,50)(17,45)(18,48)(19,47)(20,46)
(21,41)(22,44)(23,43)(24,42)(25,37)(26,40)(27,39)(28,38)(57,61)(58,64)(59,63)
(60,62)(65,81)(66,84)(67,83)(68,82)(69,77)(70,80)(71,79)(72,78)(74,76);
s2 := Sym(84)!( 1, 2)( 5, 6)( 9,10)(13,14)(17,18)(21,22)(25,26)(29,30)(33,34)
(37,38)(41,42)(45,46)(49,50)(53,54)(57,58)(61,62)(65,66)(69,70)(73,74)(77,78)
(81,82);
poly := sub<Sym(84)|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*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*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*s0*s1 >; 
 
References : None.
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