Polytope of Type {6,28}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,28}*336a
Also Known As : {6,28|2}. if this polytope has another name.
Group : SmallGroup(336,149)
Rank : 3
Schlafli Type : {6,28}
Number of vertices, edges, etc : 6, 84, 28
Order of s₀s₁s₂ : 84
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,28,2} of size 672
   {6,28,4} of size 1344
Vertex Figure Of :
   {2,6,28} of size 672
   {3,6,28} of size 1008
   {4,6,28} of size 1344
   {3,6,28} of size 1344
   {4,6,28} of size 1344
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,14}*168
   3-fold quotients : {2,28}*112
   6-fold quotients : {2,14}*56
   7-fold quotients : {6,4}*48a
   12-fold quotients : {2,7}*28
   14-fold quotients : {6,2}*24
   21-fold quotients : {2,4}*16
   28-fold quotients : {3,2}*12
   42-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,56}*672, {12,28}*672
   3-fold covers : {18,28}*1008a, {6,84}*1008a, {6,84}*1008b
   4-fold covers : {6,112}*1344, {12,28}*1344a, {24,28}*1344a, {12,56}*1344a, {24,28}*1344b, {12,56}*1344b, {6,28}*1344e
   5-fold covers : {30,28}*1680a, {6,140}*1680a
Permutation Representation (GAP) :

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

References : None.
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