Polytope of Type {2,84}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,84}*336
if this polytope has a name.
Group : SmallGroup(336,196)
Rank : 3
Schlafli Type : {2,84}
Number of vertices, edges, etc : 2, 84, 84
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,84,2} of size 672
   {2,84,4} of size 1344
   {2,84,4} of size 1344
   {2,84,4} of size 1344
Vertex Figure Of :
   {2,2,84} of size 672
   {3,2,84} of size 1008
   {4,2,84} of size 1344
   {5,2,84} of size 1680
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {2,42}*168
   3-fold quotients : {2,28}*112
   4-fold quotients : {2,21}*84
   6-fold quotients : {2,14}*56
   7-fold quotients : {2,12}*48
   12-fold quotients : {2,7}*28
   14-fold quotients : {2,6}*24
   21-fold quotients : {2,4}*16
   28-fold quotients : {2,3}*12
   42-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {4,84}*672a, {2,168}*672
   3-fold covers : {2,252}*1008, {6,84}*1008b, {6,84}*1008c
   4-fold covers : {4,168}*1344a, {4,84}*1344a, {4,168}*1344b, {8,84}*1344a, {8,84}*1344b, {2,336}*1344, {4,84}*1344b
   5-fold covers : {10,84}*1680, {2,420}*1680
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)
(25,30)(26,29)(27,28)(31,38)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)(45,66)
(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,80)(53,86)(54,85)(55,84)(56,83)
(57,82)(58,81)(59,73)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74);;
s2 := ( 3,53)( 4,52)( 5,58)( 6,57)( 7,56)( 8,55)( 9,54)(10,46)(11,45)(12,51)
(13,50)(14,49)(15,48)(16,47)(17,60)(18,59)(19,65)(20,64)(21,63)(22,62)(23,61)
(24,74)(25,73)(26,79)(27,78)(28,77)(29,76)(30,75)(31,67)(32,66)(33,72)(34,71)
(35,70)(36,69)(37,68)(38,81)(39,80)(40,86)(41,85)(42,84)(43,83)(44,82);;
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*s1*s0*s1, s0*s2*s0*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*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*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*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(86)!(1,2);
s1 := Sym(86)!( 4, 9)( 5, 8)( 6, 7)(10,17)(11,23)(12,22)(13,21)(14,20)(15,19)
(16,18)(25,30)(26,29)(27,28)(31,38)(32,44)(33,43)(34,42)(35,41)(36,40)(37,39)
(45,66)(46,72)(47,71)(48,70)(49,69)(50,68)(51,67)(52,80)(53,86)(54,85)(55,84)
(56,83)(57,82)(58,81)(59,73)(60,79)(61,78)(62,77)(63,76)(64,75)(65,74);
s2 := Sym(86)!( 3,53)( 4,52)( 5,58)( 6,57)( 7,56)( 8,55)( 9,54)(10,46)(11,45)
(12,51)(13,50)(14,49)(15,48)(16,47)(17,60)(18,59)(19,65)(20,64)(21,63)(22,62)
(23,61)(24,74)(25,73)(26,79)(27,78)(28,77)(29,76)(30,75)(31,67)(32,66)(33,72)
(34,71)(35,70)(36,69)(37,68)(38,81)(39,80)(40,86)(41,85)(42,84)(43,83)(44,82);
poly := sub<Sym(86)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s1*s0*s1, s0*s2*s0*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*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*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*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 >;

to this polytope