Polytope of Type {6,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,84}*1008d
if this polytope has a name.
Group : SmallGroup(1008,904)
Rank : 3
Schlafli Type : {6,84}
Number of vertices, edges, etc : 6, 252, 84
Order of s₀s₁s₂ : 21
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {6,28}*336b
   7-fold quotients : {6,12}*144d
   21-fold quotients : {6,4}*48b
   42-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)(31,32)(35,36)(39,40)
(43,44)(47,48)(51,52)(55,56)(59,60)(63,64)(67,68)(71,72)(75,76)(79,80)
(83,84);;
s1 := ( 2, 4)( 5,25)( 6,28)( 7,27)( 8,26)( 9,21)(10,24)(11,23)(12,22)(13,17)
(14,20)(15,19)(16,18)(29,57)(30,60)(31,59)(32,58)(33,81)(34,84)(35,83)(36,82)
(37,77)(38,80)(39,79)(40,78)(41,73)(42,76)(43,75)(44,74)(45,69)(46,72)(47,71)
(48,70)(49,65)(50,68)(51,67)(52,66)(53,61)(54,64)(55,63)(56,62);;
s2 := ( 1,34)( 2,33)( 3,36)( 4,35)( 5,30)( 6,29)( 7,32)( 8,31)( 9,54)(10,53)
(11,56)(12,55)(13,50)(14,49)(15,52)(16,51)(17,46)(18,45)(19,48)(20,47)(21,42)
(22,41)(23,44)(24,43)(25,38)(26,37)(27,40)(28,39)(57,62)(58,61)(59,64)(60,63)
(65,82)(66,81)(67,84)(68,83)(69,78)(70,77)(71,80)(72,79)(73,74)(75,76);;
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, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*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(84)!( 3, 4)( 7, 8)(11,12)(15,16)(19,20)(23,24)(27,28)(31,32)(35,36)
(39,40)(43,44)(47,48)(51,52)(55,56)(59,60)(63,64)(67,68)(71,72)(75,76)(79,80)
(83,84);
s1 := Sym(84)!( 2, 4)( 5,25)( 6,28)( 7,27)( 8,26)( 9,21)(10,24)(11,23)(12,22)
(13,17)(14,20)(15,19)(16,18)(29,57)(30,60)(31,59)(32,58)(33,81)(34,84)(35,83)
(36,82)(37,77)(38,80)(39,79)(40,78)(41,73)(42,76)(43,75)(44,74)(45,69)(46,72)
(47,71)(48,70)(49,65)(50,68)(51,67)(52,66)(53,61)(54,64)(55,63)(56,62);
s2 := Sym(84)!( 1,34)( 2,33)( 3,36)( 4,35)( 5,30)( 6,29)( 7,32)( 8,31)( 9,54)
(10,53)(11,56)(12,55)(13,50)(14,49)(15,52)(16,51)(17,46)(18,45)(19,48)(20,47)
(21,42)(22,41)(23,44)(24,43)(25,38)(26,37)(27,40)(28,39)(57,62)(58,61)(59,64)
(60,63)(65,82)(66,81)(67,84)(68,83)(69,78)(70,77)(71,80)(72,79)(73,74)(75,76);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s0*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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