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