Polytope of Type {84,6}

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

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

Permutation Representation (Magma) :

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

References : None.
to this polytope