Polytope of Type {5,2,6,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {5,2,6,14}*1680
if this polytope has a name.
Group : SmallGroup(1680,966)
Rank : 5
Schlafli Type : {5,2,6,14}
Number of vertices, edges, etc : 5, 5, 6, 42, 14
Order of s₀s₁s₂s₃s₄ : 210
Order of s₀s₁s₂s₃s₄s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   3-fold quotients : {5,2,2,14}*560
   6-fold quotients : {5,2,2,7}*280
   7-fold quotients : {5,2,6,2}*240
   14-fold quotients : {5,2,3,2}*120
   21-fold quotients : {5,2,2,2}*80
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (2,3)(4,5);;
s1 := (1,2)(3,4);;
s2 := ( 6,48)( 7,49)( 8,50)( 9,51)(10,52)(11,53)(12,54)(13,62)(14,63)(15,64)
(16,65)(17,66)(18,67)(19,68)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)(26,61)
(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,83)(35,84)(36,85)(37,86)
(38,87)(39,88)(40,89)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82);;
s3 := ( 6,55)( 7,61)( 8,60)( 9,59)(10,58)(11,57)(12,56)(13,48)(14,54)(15,53)
(16,52)(17,51)(18,50)(19,49)(20,62)(21,68)(22,67)(23,66)(24,65)(25,64)(26,63)
(27,76)(28,82)(29,81)(30,80)(31,79)(32,78)(33,77)(34,69)(35,75)(36,74)(37,73)
(38,72)(39,71)(40,70)(41,83)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84);;
s4 := ( 6,28)( 7,27)( 8,33)( 9,32)(10,31)(11,30)(12,29)(13,35)(14,34)(15,40)
(16,39)(17,38)(18,37)(19,36)(20,42)(21,41)(22,47)(23,46)(24,45)(25,44)(26,43)
(48,70)(49,69)(50,75)(51,74)(52,73)(53,72)(54,71)(55,77)(56,76)(57,82)(58,81)
(59,80)(60,79)(61,78)(62,84)(63,83)(64,89)(65,88)(66,87)(67,86)(68,85);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;  
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2, 
s1*s2*s1*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s0*s4*s0*s4, s1*s4*s1*s4, s2*s4*s2*s4, 
s2*s3*s4*s3*s2*s3*s4*s3, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(89)!(2,3)(4,5);
s1 := Sym(89)!(1,2)(3,4);
s2 := Sym(89)!( 6,48)( 7,49)( 8,50)( 9,51)(10,52)(11,53)(12,54)(13,62)(14,63)
(15,64)(16,65)(17,66)(18,67)(19,68)(20,55)(21,56)(22,57)(23,58)(24,59)(25,60)
(26,61)(27,69)(28,70)(29,71)(30,72)(31,73)(32,74)(33,75)(34,83)(35,84)(36,85)
(37,86)(38,87)(39,88)(40,89)(41,76)(42,77)(43,78)(44,79)(45,80)(46,81)(47,82);
s3 := Sym(89)!( 6,55)( 7,61)( 8,60)( 9,59)(10,58)(11,57)(12,56)(13,48)(14,54)
(15,53)(16,52)(17,51)(18,50)(19,49)(20,62)(21,68)(22,67)(23,66)(24,65)(25,64)
(26,63)(27,76)(28,82)(29,81)(30,80)(31,79)(32,78)(33,77)(34,69)(35,75)(36,74)
(37,73)(38,72)(39,71)(40,70)(41,83)(42,89)(43,88)(44,87)(45,86)(46,85)(47,84);
s4 := Sym(89)!( 6,28)( 7,27)( 8,33)( 9,32)(10,31)(11,30)(12,29)(13,35)(14,34)
(15,40)(16,39)(17,38)(18,37)(19,36)(20,42)(21,41)(22,47)(23,46)(24,45)(25,44)
(26,43)(48,70)(49,69)(50,75)(51,74)(52,73)(53,72)(54,71)(55,77)(56,76)(57,82)
(58,81)(59,80)(60,79)(61,78)(62,84)(63,83)(64,89)(65,88)(66,87)(67,86)(68,85);
poly := sub<Sym(89)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2, 
s3*s3, s4*s4, s0*s2*s0*s2, s1*s2*s1*s2, 
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4, 
s1*s4*s1*s4, s2*s4*s2*s4, s2*s3*s4*s3*s2*s3*s4*s3, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

to this polytope