Polytope of Type {28,6}

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

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

Permutation Representation (Magma) :

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

References : None.
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