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