Polytope of Type {14,10}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {14,10}*280
Also Known As : {14,10|2}. if this polytope has another name.
Group : SmallGroup(280,36)
Rank : 3
Schlafli Type : {14,10}
Number of vertices, edges, etc : 14, 70, 10
Order of s₀s₁s₂ : 70
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {14,10,2} of size 560
   {14,10,4} of size 1120
   {14,10,5} of size 1400
   {14,10,3} of size 1680
   {14,10,5} of size 1680
   {14,10,6} of size 1680
Vertex Figure Of :
   {2,14,10} of size 560
   {4,14,10} of size 1120
   {6,14,10} of size 1680
   {7,14,10} of size 1960
Quotients (Maximal Quotients in Boldface) :
   5-fold quotients : {14,2}*56
   7-fold quotients : {2,10}*40
   10-fold quotients : {7,2}*28
   14-fold quotients : {2,5}*20
   35-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {14,20}*560, {28,10}*560
   3-fold covers : {14,30}*840, {42,10}*840
   4-fold covers : {14,40}*1120, {56,10}*1120, {28,20}*1120
   5-fold covers : {14,50}*1400, {70,10}*1400a, {70,10}*1400b
   6-fold covers : {14,60}*1680, {28,30}*1680a, {42,20}*1680a, {84,10}*1680
   7-fold covers : {98,10}*1960, {14,70}*1960a, {14,70}*1960b
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)(44,49)(45,48)(46,47)
(51,56)(52,55)(53,54)(58,63)(59,62)(60,61)(65,70)(66,69)(67,68);;
s1 := ( 1, 2)( 3, 7)( 4, 6)( 8,30)( 9,29)(10,35)(11,34)(12,33)(13,32)(14,31)
(15,23)(16,22)(17,28)(18,27)(19,26)(20,25)(21,24)(36,37)(38,42)(39,41)(43,65)
(44,64)(45,70)(46,69)(47,68)(48,67)(49,66)(50,58)(51,57)(52,63)(53,62)(54,61)
(55,60)(56,59);;
s2 := ( 1,43)( 2,44)( 3,45)( 4,46)( 5,47)( 6,48)( 7,49)( 8,36)( 9,37)(10,38)
(11,39)(12,40)(13,41)(14,42)(15,64)(16,65)(17,66)(18,67)(19,68)(20,69)(21,70)
(22,57)(23,58)(24,59)(25,60)(26,61)(27,62)(28,63)(29,50)(30,51)(31,52)(32,53)
(33,54)(34,55)(35,56);;
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*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

References : None.
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