Polytope of Type {4,140}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,140}*1120
Also Known As : {4,140|2}. if this polytope has another name.
Group : SmallGroup(1120,819)
Rank : 3
Schlafli Type : {4,140}
Number of vertices, edges, etc : 4, 280, 140
Order of s₀s₁s₂ : 140
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   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) :
   2-fold quotients : {2,140}*560, {4,70}*560
   4-fold quotients : {2,70}*280
   5-fold quotients : {4,28}*224
   7-fold quotients : {4,20}*160
   8-fold quotients : {2,35}*140
   10-fold quotients : {2,28}*112, {4,14}*112
   14-fold quotients : {2,20}*80, {4,10}*80
   20-fold quotients : {2,14}*56
   28-fold quotients : {2,10}*40
   35-fold quotients : {4,4}*32
   40-fold quotients : {2,7}*28
   56-fold quotients : {2,5}*20
   70-fold quotients : {2,4}*16, {4,2}*16
   140-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (141,211)(142,212)(143,213)(144,214)(145,215)(146,216)(147,217)(148,218)
(149,219)(150,220)(151,221)(152,222)(153,223)(154,224)(155,225)(156,226)
(157,227)(158,228)(159,229)(160,230)(161,231)(162,232)(163,233)(164,234)
(165,235)(166,236)(167,237)(168,238)(169,239)(170,240)(171,241)(172,242)
(173,243)(174,244)(175,245)(176,246)(177,247)(178,248)(179,249)(180,250)
(181,251)(182,252)(183,253)(184,254)(185,255)(186,256)(187,257)(188,258)
(189,259)(190,260)(191,261)(192,262)(193,263)(194,264)(195,265)(196,266)
(197,267)(198,268)(199,269)(200,270)(201,271)(202,272)(203,273)(204,274)
(205,275)(206,276)(207,277)(208,278)(209,279)(210,280);;
s1 := (  1,141)(  2,147)(  3,146)(  4,145)(  5,144)(  6,143)(  7,142)(  8,169)
(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,162)( 16,168)
( 17,167)( 18,166)( 19,165)( 20,164)( 21,163)( 22,155)( 23,161)( 24,160)
( 25,159)( 26,158)( 27,157)( 28,156)( 29,148)( 30,154)( 31,153)( 32,152)
( 33,151)( 34,150)( 35,149)( 36,176)( 37,182)( 38,181)( 39,180)( 40,179)
( 41,178)( 42,177)( 43,204)( 44,210)( 45,209)( 46,208)( 47,207)( 48,206)
( 49,205)( 50,197)( 51,203)( 52,202)( 53,201)( 54,200)( 55,199)( 56,198)
( 57,190)( 58,196)( 59,195)( 60,194)( 61,193)( 62,192)( 63,191)( 64,183)
( 65,189)( 66,188)( 67,187)( 68,186)( 69,185)( 70,184)( 71,211)( 72,217)
( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,239)( 79,245)( 80,244)
( 81,243)( 82,242)( 83,241)( 84,240)( 85,232)( 86,238)( 87,237)( 88,236)
( 89,235)( 90,234)( 91,233)( 92,225)( 93,231)( 94,230)( 95,229)( 96,228)
( 97,227)( 98,226)( 99,218)(100,224)(101,223)(102,222)(103,221)(104,220)
(105,219)(106,246)(107,252)(108,251)(109,250)(110,249)(111,248)(112,247)
(113,274)(114,280)(115,279)(116,278)(117,277)(118,276)(119,275)(120,267)
(121,273)(122,272)(123,271)(124,270)(125,269)(126,268)(127,260)(128,266)
(129,265)(130,264)(131,263)(132,262)(133,261)(134,253)(135,259)(136,258)
(137,257)(138,256)(139,255)(140,254);;
s2 := (  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)( 15, 30)
( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)( 24, 28)
( 25, 27)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)( 41, 46)( 42, 45)
( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)( 57, 58)
( 59, 63)( 60, 62)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)( 76, 81)
( 77, 80)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)( 91,101)
( 92, 93)( 94, 98)( 95, 97)(106,114)(107,113)(108,119)(109,118)(110,117)
(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)(124,138)(125,137)
(126,136)(127,128)(129,133)(130,132)(141,184)(142,183)(143,189)(144,188)
(145,187)(146,186)(147,185)(148,177)(149,176)(150,182)(151,181)(152,180)
(153,179)(154,178)(155,205)(156,204)(157,210)(158,209)(159,208)(160,207)
(161,206)(162,198)(163,197)(164,203)(165,202)(166,201)(167,200)(168,199)
(169,191)(170,190)(171,196)(172,195)(173,194)(174,193)(175,192)(211,254)
(212,253)(213,259)(214,258)(215,257)(216,256)(217,255)(218,247)(219,246)
(220,252)(221,251)(222,250)(223,249)(224,248)(225,275)(226,274)(227,280)
(228,279)(229,278)(230,277)(231,276)(232,268)(233,267)(234,273)(235,272)
(236,271)(237,270)(238,269)(239,261)(240,260)(241,266)(242,265)(243,264)
(244,263)(245,262);;
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*s0*s1*s0*s1*s0*s1, 
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*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*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*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*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*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(280)!(141,211)(142,212)(143,213)(144,214)(145,215)(146,216)(147,217)
(148,218)(149,219)(150,220)(151,221)(152,222)(153,223)(154,224)(155,225)
(156,226)(157,227)(158,228)(159,229)(160,230)(161,231)(162,232)(163,233)
(164,234)(165,235)(166,236)(167,237)(168,238)(169,239)(170,240)(171,241)
(172,242)(173,243)(174,244)(175,245)(176,246)(177,247)(178,248)(179,249)
(180,250)(181,251)(182,252)(183,253)(184,254)(185,255)(186,256)(187,257)
(188,258)(189,259)(190,260)(191,261)(192,262)(193,263)(194,264)(195,265)
(196,266)(197,267)(198,268)(199,269)(200,270)(201,271)(202,272)(203,273)
(204,274)(205,275)(206,276)(207,277)(208,278)(209,279)(210,280);
s1 := Sym(280)!(  1,141)(  2,147)(  3,146)(  4,145)(  5,144)(  6,143)(  7,142)
(  8,169)(  9,175)( 10,174)( 11,173)( 12,172)( 13,171)( 14,170)( 15,162)
( 16,168)( 17,167)( 18,166)( 19,165)( 20,164)( 21,163)( 22,155)( 23,161)
( 24,160)( 25,159)( 26,158)( 27,157)( 28,156)( 29,148)( 30,154)( 31,153)
( 32,152)( 33,151)( 34,150)( 35,149)( 36,176)( 37,182)( 38,181)( 39,180)
( 40,179)( 41,178)( 42,177)( 43,204)( 44,210)( 45,209)( 46,208)( 47,207)
( 48,206)( 49,205)( 50,197)( 51,203)( 52,202)( 53,201)( 54,200)( 55,199)
( 56,198)( 57,190)( 58,196)( 59,195)( 60,194)( 61,193)( 62,192)( 63,191)
( 64,183)( 65,189)( 66,188)( 67,187)( 68,186)( 69,185)( 70,184)( 71,211)
( 72,217)( 73,216)( 74,215)( 75,214)( 76,213)( 77,212)( 78,239)( 79,245)
( 80,244)( 81,243)( 82,242)( 83,241)( 84,240)( 85,232)( 86,238)( 87,237)
( 88,236)( 89,235)( 90,234)( 91,233)( 92,225)( 93,231)( 94,230)( 95,229)
( 96,228)( 97,227)( 98,226)( 99,218)(100,224)(101,223)(102,222)(103,221)
(104,220)(105,219)(106,246)(107,252)(108,251)(109,250)(110,249)(111,248)
(112,247)(113,274)(114,280)(115,279)(116,278)(117,277)(118,276)(119,275)
(120,267)(121,273)(122,272)(123,271)(124,270)(125,269)(126,268)(127,260)
(128,266)(129,265)(130,264)(131,263)(132,262)(133,261)(134,253)(135,259)
(136,258)(137,257)(138,256)(139,255)(140,254);
s2 := Sym(280)!(  1,  9)(  2,  8)(  3, 14)(  4, 13)(  5, 12)(  6, 11)(  7, 10)
( 15, 30)( 16, 29)( 17, 35)( 18, 34)( 19, 33)( 20, 32)( 21, 31)( 22, 23)
( 24, 28)( 25, 27)( 36, 44)( 37, 43)( 38, 49)( 39, 48)( 40, 47)( 41, 46)
( 42, 45)( 50, 65)( 51, 64)( 52, 70)( 53, 69)( 54, 68)( 55, 67)( 56, 66)
( 57, 58)( 59, 63)( 60, 62)( 71, 79)( 72, 78)( 73, 84)( 74, 83)( 75, 82)
( 76, 81)( 77, 80)( 85,100)( 86, 99)( 87,105)( 88,104)( 89,103)( 90,102)
( 91,101)( 92, 93)( 94, 98)( 95, 97)(106,114)(107,113)(108,119)(109,118)
(110,117)(111,116)(112,115)(120,135)(121,134)(122,140)(123,139)(124,138)
(125,137)(126,136)(127,128)(129,133)(130,132)(141,184)(142,183)(143,189)
(144,188)(145,187)(146,186)(147,185)(148,177)(149,176)(150,182)(151,181)
(152,180)(153,179)(154,178)(155,205)(156,204)(157,210)(158,209)(159,208)
(160,207)(161,206)(162,198)(163,197)(164,203)(165,202)(166,201)(167,200)
(168,199)(169,191)(170,190)(171,196)(172,195)(173,194)(174,193)(175,192)
(211,254)(212,253)(213,259)(214,258)(215,257)(216,256)(217,255)(218,247)
(219,246)(220,252)(221,251)(222,250)(223,249)(224,248)(225,275)(226,274)
(227,280)(228,279)(229,278)(230,277)(231,276)(232,268)(233,267)(234,273)
(235,272)(236,271)(237,270)(238,269)(239,261)(240,260)(241,266)(242,265)
(243,264)(244,263)(245,262);
poly := sub<Sym(280)|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*s0*s1*s0*s1*s0*s1, 
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*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*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*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*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*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.
to this polytope