Polytope of Type {2,340}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,340}*1360
if this polytope has a name.
Group : SmallGroup(1360,208)
Rank : 3
Schlafli Type : {2,340}
Number of vertices, edges, etc : 2, 340, 340
Order of s₀s₁s₂ : 340
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Degenerate
   Universal
   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 : {2,68}*272
   10-fold quotients : {2,34}*136
   17-fold quotients : {2,20}*80
   20-fold quotients : {2,17}*68
   34-fold quotients : {2,10}*40
   68-fold quotients : {2,5}*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 := (1,2);;
s1 := (  4, 19)(  5, 18)(  6, 17)(  7, 16)(  8, 15)(  9, 14)( 10, 13)( 11, 12)
( 20, 71)( 21, 87)( 22, 86)( 23, 85)( 24, 84)( 25, 83)( 26, 82)( 27, 81)
( 28, 80)( 29, 79)( 30, 78)( 31, 77)( 32, 76)( 33, 75)( 34, 74)( 35, 73)
( 36, 72)( 37, 54)( 38, 70)( 39, 69)( 40, 68)( 41, 67)( 42, 66)( 43, 65)
( 44, 64)( 45, 63)( 46, 62)( 47, 61)( 48, 60)( 49, 59)( 50, 58)( 51, 57)
( 52, 56)( 53, 55)( 89,104)( 90,103)( 91,102)( 92,101)( 93,100)( 94, 99)
( 95, 98)( 96, 97)(105,156)(106,172)(107,171)(108,170)(109,169)(110,168)
(111,167)(112,166)(113,165)(114,164)(115,163)(116,162)(117,161)(118,160)
(119,159)(120,158)(121,157)(122,139)(123,155)(124,154)(125,153)(126,152)
(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)(134,144)
(135,143)(136,142)(137,141)(138,140)(173,258)(174,274)(175,273)(176,272)
(177,271)(178,270)(179,269)(180,268)(181,267)(182,266)(183,265)(184,264)
(185,263)(186,262)(187,261)(188,260)(189,259)(190,326)(191,342)(192,341)
(193,340)(194,339)(195,338)(196,337)(197,336)(198,335)(199,334)(200,333)
(201,332)(202,331)(203,330)(204,329)(205,328)(206,327)(207,309)(208,325)
(209,324)(210,323)(211,322)(212,321)(213,320)(214,319)(215,318)(216,317)
(217,316)(218,315)(219,314)(220,313)(221,312)(222,311)(223,310)(224,292)
(225,308)(226,307)(227,306)(228,305)(229,304)(230,303)(231,302)(232,301)
(233,300)(234,299)(235,298)(236,297)(237,296)(238,295)(239,294)(240,293)
(241,275)(242,291)(243,290)(244,289)(245,288)(246,287)(247,286)(248,285)
(249,284)(250,283)(251,282)(252,281)(253,280)(254,279)(255,278)(256,277)
(257,276);;
s2 := (  3,191)(  4,190)(  5,206)(  6,205)(  7,204)(  8,203)(  9,202)( 10,201)
( 11,200)( 12,199)( 13,198)( 14,197)( 15,196)( 16,195)( 17,194)( 18,193)
( 19,192)( 20,174)( 21,173)( 22,189)( 23,188)( 24,187)( 25,186)( 26,185)
( 27,184)( 28,183)( 29,182)( 30,181)( 31,180)( 32,179)( 33,178)( 34,177)
( 35,176)( 36,175)( 37,242)( 38,241)( 39,257)( 40,256)( 41,255)( 42,254)
( 43,253)( 44,252)( 45,251)( 46,250)( 47,249)( 48,248)( 49,247)( 50,246)
( 51,245)( 52,244)( 53,243)( 54,225)( 55,224)( 56,240)( 57,239)( 58,238)
( 59,237)( 60,236)( 61,235)( 62,234)( 63,233)( 64,232)( 65,231)( 66,230)
( 67,229)( 68,228)( 69,227)( 70,226)( 71,208)( 72,207)( 73,223)( 74,222)
( 75,221)( 76,220)( 77,219)( 78,218)( 79,217)( 80,216)( 81,215)( 82,214)
( 83,213)( 84,212)( 85,211)( 86,210)( 87,209)( 88,276)( 89,275)( 90,291)
( 91,290)( 92,289)( 93,288)( 94,287)( 95,286)( 96,285)( 97,284)( 98,283)
( 99,282)(100,281)(101,280)(102,279)(103,278)(104,277)(105,259)(106,258)
(107,274)(108,273)(109,272)(110,271)(111,270)(112,269)(113,268)(114,267)
(115,266)(116,265)(117,264)(118,263)(119,262)(120,261)(121,260)(122,327)
(123,326)(124,342)(125,341)(126,340)(127,339)(128,338)(129,337)(130,336)
(131,335)(132,334)(133,333)(134,332)(135,331)(136,330)(137,329)(138,328)
(139,310)(140,309)(141,325)(142,324)(143,323)(144,322)(145,321)(146,320)
(147,319)(148,318)(149,317)(150,316)(151,315)(152,314)(153,313)(154,312)
(155,311)(156,293)(157,292)(158,308)(159,307)(160,306)(161,305)(162,304)
(163,303)(164,302)(165,301)(166,300)(167,299)(168,298)(169,297)(170,296)
(171,295)(172,294);;
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*s1*s0*s1, s0*s2*s0*s2, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(342)!(1,2);
s1 := Sym(342)!(  4, 19)(  5, 18)(  6, 17)(  7, 16)(  8, 15)(  9, 14)( 10, 13)
( 11, 12)( 20, 71)( 21, 87)( 22, 86)( 23, 85)( 24, 84)( 25, 83)( 26, 82)
( 27, 81)( 28, 80)( 29, 79)( 30, 78)( 31, 77)( 32, 76)( 33, 75)( 34, 74)
( 35, 73)( 36, 72)( 37, 54)( 38, 70)( 39, 69)( 40, 68)( 41, 67)( 42, 66)
( 43, 65)( 44, 64)( 45, 63)( 46, 62)( 47, 61)( 48, 60)( 49, 59)( 50, 58)
( 51, 57)( 52, 56)( 53, 55)( 89,104)( 90,103)( 91,102)( 92,101)( 93,100)
( 94, 99)( 95, 98)( 96, 97)(105,156)(106,172)(107,171)(108,170)(109,169)
(110,168)(111,167)(112,166)(113,165)(114,164)(115,163)(116,162)(117,161)
(118,160)(119,159)(120,158)(121,157)(122,139)(123,155)(124,154)(125,153)
(126,152)(127,151)(128,150)(129,149)(130,148)(131,147)(132,146)(133,145)
(134,144)(135,143)(136,142)(137,141)(138,140)(173,258)(174,274)(175,273)
(176,272)(177,271)(178,270)(179,269)(180,268)(181,267)(182,266)(183,265)
(184,264)(185,263)(186,262)(187,261)(188,260)(189,259)(190,326)(191,342)
(192,341)(193,340)(194,339)(195,338)(196,337)(197,336)(198,335)(199,334)
(200,333)(201,332)(202,331)(203,330)(204,329)(205,328)(206,327)(207,309)
(208,325)(209,324)(210,323)(211,322)(212,321)(213,320)(214,319)(215,318)
(216,317)(217,316)(218,315)(219,314)(220,313)(221,312)(222,311)(223,310)
(224,292)(225,308)(226,307)(227,306)(228,305)(229,304)(230,303)(231,302)
(232,301)(233,300)(234,299)(235,298)(236,297)(237,296)(238,295)(239,294)
(240,293)(241,275)(242,291)(243,290)(244,289)(245,288)(246,287)(247,286)
(248,285)(249,284)(250,283)(251,282)(252,281)(253,280)(254,279)(255,278)
(256,277)(257,276);
s2 := Sym(342)!(  3,191)(  4,190)(  5,206)(  6,205)(  7,204)(  8,203)(  9,202)
( 10,201)( 11,200)( 12,199)( 13,198)( 14,197)( 15,196)( 16,195)( 17,194)
( 18,193)( 19,192)( 20,174)( 21,173)( 22,189)( 23,188)( 24,187)( 25,186)
( 26,185)( 27,184)( 28,183)( 29,182)( 30,181)( 31,180)( 32,179)( 33,178)
( 34,177)( 35,176)( 36,175)( 37,242)( 38,241)( 39,257)( 40,256)( 41,255)
( 42,254)( 43,253)( 44,252)( 45,251)( 46,250)( 47,249)( 48,248)( 49,247)
( 50,246)( 51,245)( 52,244)( 53,243)( 54,225)( 55,224)( 56,240)( 57,239)
( 58,238)( 59,237)( 60,236)( 61,235)( 62,234)( 63,233)( 64,232)( 65,231)
( 66,230)( 67,229)( 68,228)( 69,227)( 70,226)( 71,208)( 72,207)( 73,223)
( 74,222)( 75,221)( 76,220)( 77,219)( 78,218)( 79,217)( 80,216)( 81,215)
( 82,214)( 83,213)( 84,212)( 85,211)( 86,210)( 87,209)( 88,276)( 89,275)
( 90,291)( 91,290)( 92,289)( 93,288)( 94,287)( 95,286)( 96,285)( 97,284)
( 98,283)( 99,282)(100,281)(101,280)(102,279)(103,278)(104,277)(105,259)
(106,258)(107,274)(108,273)(109,272)(110,271)(111,270)(112,269)(113,268)
(114,267)(115,266)(116,265)(117,264)(118,263)(119,262)(120,261)(121,260)
(122,327)(123,326)(124,342)(125,341)(126,340)(127,339)(128,338)(129,337)
(130,336)(131,335)(132,334)(133,333)(134,332)(135,331)(136,330)(137,329)
(138,328)(139,310)(140,309)(141,325)(142,324)(143,323)(144,322)(145,321)
(146,320)(147,319)(148,318)(149,317)(150,316)(151,315)(152,314)(153,313)
(154,312)(155,311)(156,293)(157,292)(158,308)(159,307)(160,306)(161,305)
(162,304)(163,303)(164,302)(165,301)(166,300)(167,299)(168,298)(169,297)
(170,296)(171,295)(172,294);
poly := sub<Sym(342)|s0,s1,s2>;

Finitely Presented Group Representation (Magma) :

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

to this polytope