Polytope of Type {84,2}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,2}*336
if this polytope has a name.
Group : SmallGroup(336,196)
Rank : 3
Schlafli Type : {84,2}
Number of vertices, edges, etc : 84, 84, 2
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {84,2,2} of size 672
   {84,2,3} of size 1008
   {84,2,4} of size 1344
   {84,2,5} of size 1680
Vertex Figure Of :
   {2,84,2} of size 672
   {4,84,2} of size 1344
   {4,84,2} of size 1344
   {4,84,2} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {42,2}*168
   3-fold quotients : {28,2}*112
   4-fold quotients : {21,2}*84
   6-fold quotients : {14,2}*56
   7-fold quotients : {12,2}*48
   12-fold quotients : {7,2}*28
   14-fold quotients : {6,2}*24
   21-fold quotients : {4,2}*16
   28-fold quotients : {3,2}*12
   42-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {84,4}*672a, {168,2}*672
   3-fold covers : {252,2}*1008, {84,6}*1008b, {84,6}*1008c
   4-fold covers : {168,4}*1344a, {84,4}*1344a, {168,4}*1344b, {84,8}*1344a, {84,8}*1344b, {336,2}*1344, {84,4}*1344b
   5-fold covers : {84,10}*1680, {420,2}*1680
Permutation Representation (GAP) :

s0 := ( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)(14,16)
(23,28)(24,27)(25,26)(29,36)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)(43,64)
(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,78)(51,84)(52,83)(53,82)(54,81)
(55,80)(56,79)(57,71)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72);;
s1 := ( 1,51)( 2,50)( 3,56)( 4,55)( 5,54)( 6,53)( 7,52)( 8,44)( 9,43)(10,49)
(11,48)(12,47)(13,46)(14,45)(15,58)(16,57)(17,63)(18,62)(19,61)(20,60)(21,59)
(22,72)(23,71)(24,77)(25,76)(26,75)(27,74)(28,73)(29,65)(30,64)(31,70)(32,69)
(33,68)(34,67)(35,66)(36,79)(37,78)(38,84)(39,83)(40,82)(41,81)(42,80);;
s2 := (85,86);;
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, 
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*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*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*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(86)!( 2, 7)( 3, 6)( 4, 5)( 8,15)( 9,21)(10,20)(11,19)(12,18)(13,17)
(14,16)(23,28)(24,27)(25,26)(29,36)(30,42)(31,41)(32,40)(33,39)(34,38)(35,37)
(43,64)(44,70)(45,69)(46,68)(47,67)(48,66)(49,65)(50,78)(51,84)(52,83)(53,82)
(54,81)(55,80)(56,79)(57,71)(58,77)(59,76)(60,75)(61,74)(62,73)(63,72);
s1 := Sym(86)!( 1,51)( 2,50)( 3,56)( 4,55)( 5,54)( 6,53)( 7,52)( 8,44)( 9,43)
(10,49)(11,48)(12,47)(13,46)(14,45)(15,58)(16,57)(17,63)(18,62)(19,61)(20,60)
(21,59)(22,72)(23,71)(24,77)(25,76)(26,75)(27,74)(28,73)(29,65)(30,64)(31,70)
(32,69)(33,68)(34,67)(35,66)(36,79)(37,78)(38,84)(39,83)(40,82)(41,81)(42,80);
s2 := Sym(86)!(85,86);
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*s2*s0*s2, s1*s2*s1*s2, 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*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*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*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 >;

to this polytope