Polytope of Type {2,10,14}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,10,14}*560
if this polytope has a name.
Group : SmallGroup(560,176)
Rank : 4
Schlafli Type : {2,10,14}
Number of vertices, edges, etc : 2, 10, 70, 14
Order of s₀s₁s₂s₃ : 70
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {2,10,14,2} of size 1120
Vertex Figure Of :
   {2,2,10,14} of size 1120
   {3,2,10,14} of size 1680
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {2,2,14}*112
   7-fold quotients : {2,10,2}*80
   10-fold quotients : {2,2,7}*56
   14-fold quotients : {2,5,2}*40
   35-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   2-fold covers : {2,20,14}*1120, {2,10,28}*1120, {4,10,14}*1120
   3-fold covers : {6,10,14}*1680, {2,30,14}*1680, {2,10,42}*1680
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

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

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