Polytope of Type {170,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {170,4}*1360
Also Known As : {170,4|2}. if this polytope has another name.
Group : SmallGroup(1360,210)
Rank : 3
Schlafli Type : {170,4}
Number of vertices, edges, etc : 170, 340, 4
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 : {170,2}*680
   4-fold quotients : {85,2}*340
   5-fold quotients : {34,4}*272
   10-fold quotients : {34,2}*136
   17-fold quotients : {10,4}*80
   20-fold quotients : {17,2}*68
   34-fold quotients : {10,2}*40
   68-fold quotients : {5,2}*20
   85-fold quotients : {2,4}*16
   170-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  2, 17)(  3, 16)(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)(  9, 10)
( 18, 69)( 19, 85)( 20, 84)( 21, 83)( 22, 82)( 23, 81)( 24, 80)( 25, 79)
( 26, 78)( 27, 77)( 28, 76)( 29, 75)( 30, 74)( 31, 73)( 32, 72)( 33, 71)
( 34, 70)( 35, 52)( 36, 68)( 37, 67)( 38, 66)( 39, 65)( 40, 64)( 41, 63)
( 42, 62)( 43, 61)( 44, 60)( 45, 59)( 46, 58)( 47, 57)( 48, 56)( 49, 55)
( 50, 54)( 51, 53)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)( 92, 97)
( 93, 96)( 94, 95)(103,154)(104,170)(105,169)(106,168)(107,167)(108,166)
(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)(116,158)
(117,157)(118,156)(119,155)(120,137)(121,153)(122,152)(123,151)(124,150)
(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)(132,142)
(133,141)(134,140)(135,139)(136,138)(172,187)(173,186)(174,185)(175,184)
(176,183)(177,182)(178,181)(179,180)(188,239)(189,255)(190,254)(191,253)
(192,252)(193,251)(194,250)(195,249)(196,248)(197,247)(198,246)(199,245)
(200,244)(201,243)(202,242)(203,241)(204,240)(205,222)(206,238)(207,237)
(208,236)(209,235)(210,234)(211,233)(212,232)(213,231)(214,230)(215,229)
(216,228)(217,227)(218,226)(219,225)(220,224)(221,223)(257,272)(258,271)
(259,270)(260,269)(261,268)(262,267)(263,266)(264,265)(273,324)(274,340)
(275,339)(276,338)(277,337)(278,336)(279,335)(280,334)(281,333)(282,332)
(283,331)(284,330)(285,329)(286,328)(287,327)(288,326)(289,325)(290,307)
(291,323)(292,322)(293,321)(294,320)(295,319)(296,318)(297,317)(298,316)
(299,315)(300,314)(301,313)(302,312)(303,311)(304,310)(305,309)(306,308);;
s1 := (  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)(171,274)(172,273)(173,289)(174,288)
(175,287)(176,286)(177,285)(178,284)(179,283)(180,282)(181,281)(182,280)
(183,279)(184,278)(185,277)(186,276)(187,275)(188,257)(189,256)(190,272)
(191,271)(192,270)(193,269)(194,268)(195,267)(196,266)(197,265)(198,264)
(199,263)(200,262)(201,261)(202,260)(203,259)(204,258)(205,325)(206,324)
(207,340)(208,339)(209,338)(210,337)(211,336)(212,335)(213,334)(214,333)
(215,332)(216,331)(217,330)(218,329)(219,328)(220,327)(221,326)(222,308)
(223,307)(224,323)(225,322)(226,321)(227,320)(228,319)(229,318)(230,317)
(231,316)(232,315)(233,314)(234,313)(235,312)(236,311)(237,310)(238,309)
(239,291)(240,290)(241,306)(242,305)(243,304)(244,303)(245,302)(246,301)
(247,300)(248,299)(249,298)(250,297)(251,296)(252,295)(253,294)(254,293)
(255,292);;
s2 := (  1,171)(  2,172)(  3,173)(  4,174)(  5,175)(  6,176)(  7,177)(  8,178)
(  9,179)( 10,180)( 11,181)( 12,182)( 13,183)( 14,184)( 15,185)( 16,186)
( 17,187)( 18,188)( 19,189)( 20,190)( 21,191)( 22,192)( 23,193)( 24,194)
( 25,195)( 26,196)( 27,197)( 28,198)( 29,199)( 30,200)( 31,201)( 32,202)
( 33,203)( 34,204)( 35,205)( 36,206)( 37,207)( 38,208)( 39,209)( 40,210)
( 41,211)( 42,212)( 43,213)( 44,214)( 45,215)( 46,216)( 47,217)( 48,218)
( 49,219)( 50,220)( 51,221)( 52,222)( 53,223)( 54,224)( 55,225)( 56,226)
( 57,227)( 58,228)( 59,229)( 60,230)( 61,231)( 62,232)( 63,233)( 64,234)
( 65,235)( 66,236)( 67,237)( 68,238)( 69,239)( 70,240)( 71,241)( 72,242)
( 73,243)( 74,244)( 75,245)( 76,246)( 77,247)( 78,248)( 79,249)( 80,250)
( 81,251)( 82,252)( 83,253)( 84,254)( 85,255)( 86,256)( 87,257)( 88,258)
( 89,259)( 90,260)( 91,261)( 92,262)( 93,263)( 94,264)( 95,265)( 96,266)
( 97,267)( 98,268)( 99,269)(100,270)(101,271)(102,272)(103,273)(104,274)
(105,275)(106,276)(107,277)(108,278)(109,279)(110,280)(111,281)(112,282)
(113,283)(114,284)(115,285)(116,286)(117,287)(118,288)(119,289)(120,290)
(121,291)(122,292)(123,293)(124,294)(125,295)(126,296)(127,297)(128,298)
(129,299)(130,300)(131,301)(132,302)(133,303)(134,304)(135,305)(136,306)
(137,307)(138,308)(139,309)(140,310)(141,311)(142,312)(143,313)(144,314)
(145,315)(146,316)(147,317)(148,318)(149,319)(150,320)(151,321)(152,322)
(153,323)(154,324)(155,325)(156,326)(157,327)(158,328)(159,329)(160,330)
(161,331)(162,332)(163,333)(164,334)(165,335)(166,336)(167,337)(168,338)
(169,339)(170,340);;
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, 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*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*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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(340)!(  2, 17)(  3, 16)(  4, 15)(  5, 14)(  6, 13)(  7, 12)(  8, 11)
(  9, 10)( 18, 69)( 19, 85)( 20, 84)( 21, 83)( 22, 82)( 23, 81)( 24, 80)
( 25, 79)( 26, 78)( 27, 77)( 28, 76)( 29, 75)( 30, 74)( 31, 73)( 32, 72)
( 33, 71)( 34, 70)( 35, 52)( 36, 68)( 37, 67)( 38, 66)( 39, 65)( 40, 64)
( 41, 63)( 42, 62)( 43, 61)( 44, 60)( 45, 59)( 46, 58)( 47, 57)( 48, 56)
( 49, 55)( 50, 54)( 51, 53)( 87,102)( 88,101)( 89,100)( 90, 99)( 91, 98)
( 92, 97)( 93, 96)( 94, 95)(103,154)(104,170)(105,169)(106,168)(107,167)
(108,166)(109,165)(110,164)(111,163)(112,162)(113,161)(114,160)(115,159)
(116,158)(117,157)(118,156)(119,155)(120,137)(121,153)(122,152)(123,151)
(124,150)(125,149)(126,148)(127,147)(128,146)(129,145)(130,144)(131,143)
(132,142)(133,141)(134,140)(135,139)(136,138)(172,187)(173,186)(174,185)
(175,184)(176,183)(177,182)(178,181)(179,180)(188,239)(189,255)(190,254)
(191,253)(192,252)(193,251)(194,250)(195,249)(196,248)(197,247)(198,246)
(199,245)(200,244)(201,243)(202,242)(203,241)(204,240)(205,222)(206,238)
(207,237)(208,236)(209,235)(210,234)(211,233)(212,232)(213,231)(214,230)
(215,229)(216,228)(217,227)(218,226)(219,225)(220,224)(221,223)(257,272)
(258,271)(259,270)(260,269)(261,268)(262,267)(263,266)(264,265)(273,324)
(274,340)(275,339)(276,338)(277,337)(278,336)(279,335)(280,334)(281,333)
(282,332)(283,331)(284,330)(285,329)(286,328)(287,327)(288,326)(289,325)
(290,307)(291,323)(292,322)(293,321)(294,320)(295,319)(296,318)(297,317)
(298,316)(299,315)(300,314)(301,313)(302,312)(303,311)(304,310)(305,309)
(306,308);
s1 := 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)(171,274)(172,273)(173,289)
(174,288)(175,287)(176,286)(177,285)(178,284)(179,283)(180,282)(181,281)
(182,280)(183,279)(184,278)(185,277)(186,276)(187,275)(188,257)(189,256)
(190,272)(191,271)(192,270)(193,269)(194,268)(195,267)(196,266)(197,265)
(198,264)(199,263)(200,262)(201,261)(202,260)(203,259)(204,258)(205,325)
(206,324)(207,340)(208,339)(209,338)(210,337)(211,336)(212,335)(213,334)
(214,333)(215,332)(216,331)(217,330)(218,329)(219,328)(220,327)(221,326)
(222,308)(223,307)(224,323)(225,322)(226,321)(227,320)(228,319)(229,318)
(230,317)(231,316)(232,315)(233,314)(234,313)(235,312)(236,311)(237,310)
(238,309)(239,291)(240,290)(241,306)(242,305)(243,304)(244,303)(245,302)
(246,301)(247,300)(248,299)(249,298)(250,297)(251,296)(252,295)(253,294)
(254,293)(255,292);
s2 := Sym(340)!(  1,171)(  2,172)(  3,173)(  4,174)(  5,175)(  6,176)(  7,177)
(  8,178)(  9,179)( 10,180)( 11,181)( 12,182)( 13,183)( 14,184)( 15,185)
( 16,186)( 17,187)( 18,188)( 19,189)( 20,190)( 21,191)( 22,192)( 23,193)
( 24,194)( 25,195)( 26,196)( 27,197)( 28,198)( 29,199)( 30,200)( 31,201)
( 32,202)( 33,203)( 34,204)( 35,205)( 36,206)( 37,207)( 38,208)( 39,209)
( 40,210)( 41,211)( 42,212)( 43,213)( 44,214)( 45,215)( 46,216)( 47,217)
( 48,218)( 49,219)( 50,220)( 51,221)( 52,222)( 53,223)( 54,224)( 55,225)
( 56,226)( 57,227)( 58,228)( 59,229)( 60,230)( 61,231)( 62,232)( 63,233)
( 64,234)( 65,235)( 66,236)( 67,237)( 68,238)( 69,239)( 70,240)( 71,241)
( 72,242)( 73,243)( 74,244)( 75,245)( 76,246)( 77,247)( 78,248)( 79,249)
( 80,250)( 81,251)( 82,252)( 83,253)( 84,254)( 85,255)( 86,256)( 87,257)
( 88,258)( 89,259)( 90,260)( 91,261)( 92,262)( 93,263)( 94,264)( 95,265)
( 96,266)( 97,267)( 98,268)( 99,269)(100,270)(101,271)(102,272)(103,273)
(104,274)(105,275)(106,276)(107,277)(108,278)(109,279)(110,280)(111,281)
(112,282)(113,283)(114,284)(115,285)(116,286)(117,287)(118,288)(119,289)
(120,290)(121,291)(122,292)(123,293)(124,294)(125,295)(126,296)(127,297)
(128,298)(129,299)(130,300)(131,301)(132,302)(133,303)(134,304)(135,305)
(136,306)(137,307)(138,308)(139,309)(140,310)(141,311)(142,312)(143,313)
(144,314)(145,315)(146,316)(147,317)(148,318)(149,319)(150,320)(151,321)
(152,322)(153,323)(154,324)(155,325)(156,326)(157,327)(158,328)(159,329)
(160,330)(161,331)(162,332)(163,333)(164,334)(165,335)(166,336)(167,337)
(168,338)(169,339)(170,340);
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*s2*s1*s0*s1*s2*s1, 
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*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*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*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*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*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*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*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*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*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*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*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*s0*s1*s0*s1 >;

References : None.
to this polytope