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Polytope of Type {4,170}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,170}*1360
Also Known As : {4,170|2}. if this polytope has another name.
Group : SmallGroup(1360,210)
Rank : 3
Schlafli Type : {4,170}
Number of vertices, edges, etc : 4, 340, 170
Order of s0s1s2 : 340
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 : {2,170}*680
   4-fold quotients : {2,85}*340
   5-fold quotients : {4,34}*272
   10-fold quotients : {2,34}*136
   17-fold quotients : {4,10}*80
   20-fold quotients : {2,17}*68
   34-fold quotients : {2,10}*40
   68-fold quotients : {2,5}*20
   85-fold quotients : {4,2}*16
   170-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (171,256)(172,257)(173,258)(174,259)(175,260)(176,261)(177,262)(178,263)
(179,264)(180,265)(181,266)(182,267)(183,268)(184,269)(185,270)(186,271)
(187,272)(188,273)(189,274)(190,275)(191,276)(192,277)(193,278)(194,279)
(195,280)(196,281)(197,282)(198,283)(199,284)(200,285)(201,286)(202,287)
(203,288)(204,289)(205,290)(206,291)(207,292)(208,293)(209,294)(210,295)
(211,296)(212,297)(213,298)(214,299)(215,300)(216,301)(217,302)(218,303)
(219,304)(220,305)(221,306)(222,307)(223,308)(224,309)(225,310)(226,311)
(227,312)(228,313)(229,314)(230,315)(231,316)(232,317)(233,318)(234,319)
(235,320)(236,321)(237,322)(238,323)(239,324)(240,325)(241,326)(242,327)
(243,328)(244,329)(245,330)(246,331)(247,332)(248,333)(249,334)(250,335)
(251,336)(252,337)(253,338)(254,339)(255,340);;
s1 := (  1,171)(  2,187)(  3,186)(  4,185)(  5,184)(  6,183)(  7,182)(  8,181)
(  9,180)( 10,179)( 11,178)( 12,177)( 13,176)( 14,175)( 15,174)( 16,173)
( 17,172)( 18,239)( 19,255)( 20,254)( 21,253)( 22,252)( 23,251)( 24,250)
( 25,249)( 26,248)( 27,247)( 28,246)( 29,245)( 30,244)( 31,243)( 32,242)
( 33,241)( 34,240)( 35,222)( 36,238)( 37,237)( 38,236)( 39,235)( 40,234)
( 41,233)( 42,232)( 43,231)( 44,230)( 45,229)( 46,228)( 47,227)( 48,226)
( 49,225)( 50,224)( 51,223)( 52,205)( 53,221)( 54,220)( 55,219)( 56,218)
( 57,217)( 58,216)( 59,215)( 60,214)( 61,213)( 62,212)( 63,211)( 64,210)
( 65,209)( 66,208)( 67,207)( 68,206)( 69,188)( 70,204)( 71,203)( 72,202)
( 73,201)( 74,200)( 75,199)( 76,198)( 77,197)( 78,196)( 79,195)( 80,194)
( 81,193)( 82,192)( 83,191)( 84,190)( 85,189)( 86,256)( 87,272)( 88,271)
( 89,270)( 90,269)( 91,268)( 92,267)( 93,266)( 94,265)( 95,264)( 96,263)
( 97,262)( 98,261)( 99,260)(100,259)(101,258)(102,257)(103,324)(104,340)
(105,339)(106,338)(107,337)(108,336)(109,335)(110,334)(111,333)(112,332)
(113,331)(114,330)(115,329)(116,328)(117,327)(118,326)(119,325)(120,307)
(121,323)(122,322)(123,321)(124,320)(125,319)(126,318)(127,317)(128,316)
(129,315)(130,314)(131,313)(132,312)(133,311)(134,310)(135,309)(136,308)
(137,290)(138,306)(139,305)(140,304)(141,303)(142,302)(143,301)(144,300)
(145,299)(146,298)(147,297)(148,296)(149,295)(150,294)(151,293)(152,292)
(153,291)(154,273)(155,289)(156,288)(157,287)(158,286)(159,285)(160,284)
(161,283)(162,282)(163,281)(164,280)(165,279)(166,278)(167,277)(168,276)
(169,275)(170,274);;
s2 := (  1, 19)(  2, 18)(  3, 34)(  4, 33)(  5, 32)(  6, 31)(  7, 30)(  8, 29)
(  9, 28)( 10, 27)( 11, 26)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 16, 21)
( 17, 20)( 35, 70)( 36, 69)( 37, 85)( 38, 84)( 39, 83)( 40, 82)( 41, 81)
( 42, 80)( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)( 48, 74)( 49, 73)
( 50, 72)( 51, 71)( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)( 58, 64)
( 59, 63)( 60, 62)( 86,104)( 87,103)( 88,119)( 89,118)( 90,117)( 91,116)
( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)( 99,108)
(100,107)(101,106)(102,105)(120,155)(121,154)(122,170)(123,169)(124,168)
(125,167)(126,166)(127,165)(128,164)(129,163)(130,162)(131,161)(132,160)
(133,159)(134,158)(135,157)(136,156)(137,138)(139,153)(140,152)(141,151)
(142,150)(143,149)(144,148)(145,147)(171,189)(172,188)(173,204)(174,203)
(175,202)(176,201)(177,200)(178,199)(179,198)(180,197)(181,196)(182,195)
(183,194)(184,193)(185,192)(186,191)(187,190)(205,240)(206,239)(207,255)
(208,254)(209,253)(210,252)(211,251)(212,250)(213,249)(214,248)(215,247)
(216,246)(217,245)(218,244)(219,243)(220,242)(221,241)(222,223)(224,238)
(225,237)(226,236)(227,235)(228,234)(229,233)(230,232)(256,274)(257,273)
(258,289)(259,288)(260,287)(261,286)(262,285)(263,284)(264,283)(265,282)
(266,281)(267,280)(268,279)(269,278)(270,277)(271,276)(272,275)(290,325)
(291,324)(292,340)(293,339)(294,338)(295,337)(296,336)(297,335)(298,334)
(299,333)(300,332)(301,331)(302,330)(303,329)(304,328)(305,327)(306,326)
(307,308)(309,323)(310,322)(311,321)(312,320)(313,319)(314,318)(315,317);;
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*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(340)!(171,256)(172,257)(173,258)(174,259)(175,260)(176,261)(177,262)
(178,263)(179,264)(180,265)(181,266)(182,267)(183,268)(184,269)(185,270)
(186,271)(187,272)(188,273)(189,274)(190,275)(191,276)(192,277)(193,278)
(194,279)(195,280)(196,281)(197,282)(198,283)(199,284)(200,285)(201,286)
(202,287)(203,288)(204,289)(205,290)(206,291)(207,292)(208,293)(209,294)
(210,295)(211,296)(212,297)(213,298)(214,299)(215,300)(216,301)(217,302)
(218,303)(219,304)(220,305)(221,306)(222,307)(223,308)(224,309)(225,310)
(226,311)(227,312)(228,313)(229,314)(230,315)(231,316)(232,317)(233,318)
(234,319)(235,320)(236,321)(237,322)(238,323)(239,324)(240,325)(241,326)
(242,327)(243,328)(244,329)(245,330)(246,331)(247,332)(248,333)(249,334)
(250,335)(251,336)(252,337)(253,338)(254,339)(255,340);
s1 := Sym(340)!(  1,171)(  2,187)(  3,186)(  4,185)(  5,184)(  6,183)(  7,182)
(  8,181)(  9,180)( 10,179)( 11,178)( 12,177)( 13,176)( 14,175)( 15,174)
( 16,173)( 17,172)( 18,239)( 19,255)( 20,254)( 21,253)( 22,252)( 23,251)
( 24,250)( 25,249)( 26,248)( 27,247)( 28,246)( 29,245)( 30,244)( 31,243)
( 32,242)( 33,241)( 34,240)( 35,222)( 36,238)( 37,237)( 38,236)( 39,235)
( 40,234)( 41,233)( 42,232)( 43,231)( 44,230)( 45,229)( 46,228)( 47,227)
( 48,226)( 49,225)( 50,224)( 51,223)( 52,205)( 53,221)( 54,220)( 55,219)
( 56,218)( 57,217)( 58,216)( 59,215)( 60,214)( 61,213)( 62,212)( 63,211)
( 64,210)( 65,209)( 66,208)( 67,207)( 68,206)( 69,188)( 70,204)( 71,203)
( 72,202)( 73,201)( 74,200)( 75,199)( 76,198)( 77,197)( 78,196)( 79,195)
( 80,194)( 81,193)( 82,192)( 83,191)( 84,190)( 85,189)( 86,256)( 87,272)
( 88,271)( 89,270)( 90,269)( 91,268)( 92,267)( 93,266)( 94,265)( 95,264)
( 96,263)( 97,262)( 98,261)( 99,260)(100,259)(101,258)(102,257)(103,324)
(104,340)(105,339)(106,338)(107,337)(108,336)(109,335)(110,334)(111,333)
(112,332)(113,331)(114,330)(115,329)(116,328)(117,327)(118,326)(119,325)
(120,307)(121,323)(122,322)(123,321)(124,320)(125,319)(126,318)(127,317)
(128,316)(129,315)(130,314)(131,313)(132,312)(133,311)(134,310)(135,309)
(136,308)(137,290)(138,306)(139,305)(140,304)(141,303)(142,302)(143,301)
(144,300)(145,299)(146,298)(147,297)(148,296)(149,295)(150,294)(151,293)
(152,292)(153,291)(154,273)(155,289)(156,288)(157,287)(158,286)(159,285)
(160,284)(161,283)(162,282)(163,281)(164,280)(165,279)(166,278)(167,277)
(168,276)(169,275)(170,274);
s2 := Sym(340)!(  1, 19)(  2, 18)(  3, 34)(  4, 33)(  5, 32)(  6, 31)(  7, 30)
(  8, 29)(  9, 28)( 10, 27)( 11, 26)( 12, 25)( 13, 24)( 14, 23)( 15, 22)
( 16, 21)( 17, 20)( 35, 70)( 36, 69)( 37, 85)( 38, 84)( 39, 83)( 40, 82)
( 41, 81)( 42, 80)( 43, 79)( 44, 78)( 45, 77)( 46, 76)( 47, 75)( 48, 74)
( 49, 73)( 50, 72)( 51, 71)( 52, 53)( 54, 68)( 55, 67)( 56, 66)( 57, 65)
( 58, 64)( 59, 63)( 60, 62)( 86,104)( 87,103)( 88,119)( 89,118)( 90,117)
( 91,116)( 92,115)( 93,114)( 94,113)( 95,112)( 96,111)( 97,110)( 98,109)
( 99,108)(100,107)(101,106)(102,105)(120,155)(121,154)(122,170)(123,169)
(124,168)(125,167)(126,166)(127,165)(128,164)(129,163)(130,162)(131,161)
(132,160)(133,159)(134,158)(135,157)(136,156)(137,138)(139,153)(140,152)
(141,151)(142,150)(143,149)(144,148)(145,147)(171,189)(172,188)(173,204)
(174,203)(175,202)(176,201)(177,200)(178,199)(179,198)(180,197)(181,196)
(182,195)(183,194)(184,193)(185,192)(186,191)(187,190)(205,240)(206,239)
(207,255)(208,254)(209,253)(210,252)(211,251)(212,250)(213,249)(214,248)
(215,247)(216,246)(217,245)(218,244)(219,243)(220,242)(221,241)(222,223)
(224,238)(225,237)(226,236)(227,235)(228,234)(229,233)(230,232)(256,274)
(257,273)(258,289)(259,288)(260,287)(261,286)(262,285)(263,284)(264,283)
(265,282)(266,281)(267,280)(268,279)(269,278)(270,277)(271,276)(272,275)
(290,325)(291,324)(292,340)(293,339)(294,338)(295,337)(296,336)(297,335)
(298,334)(299,333)(300,332)(301,331)(302,330)(303,329)(304,328)(305,327)
(306,326)(307,308)(309,323)(310,322)(311,321)(312,320)(313,319)(314,318)
(315,317);
poly := sub<Sym(340)|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*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