Polytope of Type {6,110}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,110}*1320
Also Known As : {6,110|2}. if this polytope has another name.
Group : SmallGroup(1320,171)
Rank : 3
Schlafli Type : {6,110}
Number of vertices, edges, etc : 6, 330, 110
Order of s0s1s2 : 330
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) :
3-fold quotients : {2,110}*440
5-fold quotients : {6,22}*264
6-fold quotients : {2,55}*220
11-fold quotients : {6,10}*120
15-fold quotients : {2,22}*88
30-fold quotients : {2,11}*44
33-fold quotients : {2,10}*40
55-fold quotients : {6,2}*24
66-fold quotients : {2,5}*20
110-fold quotients : {3,2}*12
165-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 56,111)( 57,112)( 58,113)( 59,114)( 60,115)( 61,116)( 62,117)( 63,118)
( 64,119)( 65,120)( 66,121)( 67,122)( 68,123)( 69,124)( 70,125)( 71,126)
( 72,127)( 73,128)( 74,129)( 75,130)( 76,131)( 77,132)( 78,133)( 79,134)
( 80,135)( 81,136)( 82,137)( 83,138)( 84,139)( 85,140)( 86,141)( 87,142)
( 88,143)( 89,144)( 90,145)( 91,146)( 92,147)( 93,148)( 94,149)( 95,150)
( 96,151)( 97,152)( 98,153)( 99,154)(100,155)(101,156)(102,157)(103,158)
(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)(221,276)
(222,277)(223,278)(224,279)(225,280)(226,281)(227,282)(228,283)(229,284)
(230,285)(231,286)(232,287)(233,288)(234,289)(235,290)(236,291)(237,292)
(238,293)(239,294)(240,295)(241,296)(242,297)(243,298)(244,299)(245,300)
(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)(253,308)
(254,309)(255,310)(256,311)(257,312)(258,313)(259,314)(260,315)(261,316)
(262,317)(263,318)(264,319)(265,320)(266,321)(267,322)(268,323)(269,324)
(270,325)(271,326)(272,327)(273,328)(274,329)(275,330);;
s1 := ( 1, 56)( 2, 66)( 3, 65)( 4, 64)( 5, 63)( 6, 62)( 7, 61)( 8, 60)
( 9, 59)( 10, 58)( 11, 57)( 12,100)( 13,110)( 14,109)( 15,108)( 16,107)
( 17,106)( 18,105)( 19,104)( 20,103)( 21,102)( 22,101)( 23, 89)( 24, 99)
( 25, 98)( 26, 97)( 27, 96)( 28, 95)( 29, 94)( 30, 93)( 31, 92)( 32, 91)
( 33, 90)( 34, 78)( 35, 88)( 36, 87)( 37, 86)( 38, 85)( 39, 84)( 40, 83)
( 41, 82)( 42, 81)( 43, 80)( 44, 79)( 45, 67)( 46, 77)( 47, 76)( 48, 75)
( 49, 74)( 50, 73)( 51, 72)( 52, 71)( 53, 70)( 54, 69)( 55, 68)(112,121)
(113,120)(114,119)(115,118)(116,117)(122,155)(123,165)(124,164)(125,163)
(126,162)(127,161)(128,160)(129,159)(130,158)(131,157)(132,156)(133,144)
(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)
(142,146)(143,145)(166,221)(167,231)(168,230)(169,229)(170,228)(171,227)
(172,226)(173,225)(174,224)(175,223)(176,222)(177,265)(178,275)(179,274)
(180,273)(181,272)(182,271)(183,270)(184,269)(185,268)(186,267)(187,266)
(188,254)(189,264)(190,263)(191,262)(192,261)(193,260)(194,259)(195,258)
(196,257)(197,256)(198,255)(199,243)(200,253)(201,252)(202,251)(203,250)
(204,249)(205,248)(206,247)(207,246)(208,245)(209,244)(210,232)(211,242)
(212,241)(213,240)(214,239)(215,238)(216,237)(217,236)(218,235)(219,234)
(220,233)(277,286)(278,285)(279,284)(280,283)(281,282)(287,320)(288,330)
(289,329)(290,328)(291,327)(292,326)(293,325)(294,324)(295,323)(296,322)
(297,321)(298,309)(299,319)(300,318)(301,317)(302,316)(303,315)(304,314)
(305,313)(306,312)(307,311)(308,310);;
s2 := ( 1,178)( 2,177)( 3,187)( 4,186)( 5,185)( 6,184)( 7,183)( 8,182)
( 9,181)( 10,180)( 11,179)( 12,167)( 13,166)( 14,176)( 15,175)( 16,174)
( 17,173)( 18,172)( 19,171)( 20,170)( 21,169)( 22,168)( 23,211)( 24,210)
( 25,220)( 26,219)( 27,218)( 28,217)( 29,216)( 30,215)( 31,214)( 32,213)
( 33,212)( 34,200)( 35,199)( 36,209)( 37,208)( 38,207)( 39,206)( 40,205)
( 41,204)( 42,203)( 43,202)( 44,201)( 45,189)( 46,188)( 47,198)( 48,197)
( 49,196)( 50,195)( 51,194)( 52,193)( 53,192)( 54,191)( 55,190)( 56,233)
( 57,232)( 58,242)( 59,241)( 60,240)( 61,239)( 62,238)( 63,237)( 64,236)
( 65,235)( 66,234)( 67,222)( 68,221)( 69,231)( 70,230)( 71,229)( 72,228)
( 73,227)( 74,226)( 75,225)( 76,224)( 77,223)( 78,266)( 79,265)( 80,275)
( 81,274)( 82,273)( 83,272)( 84,271)( 85,270)( 86,269)( 87,268)( 88,267)
( 89,255)( 90,254)( 91,264)( 92,263)( 93,262)( 94,261)( 95,260)( 96,259)
( 97,258)( 98,257)( 99,256)(100,244)(101,243)(102,253)(103,252)(104,251)
(105,250)(106,249)(107,248)(108,247)(109,246)(110,245)(111,288)(112,287)
(113,297)(114,296)(115,295)(116,294)(117,293)(118,292)(119,291)(120,290)
(121,289)(122,277)(123,276)(124,286)(125,285)(126,284)(127,283)(128,282)
(129,281)(130,280)(131,279)(132,278)(133,321)(134,320)(135,330)(136,329)
(137,328)(138,327)(139,326)(140,325)(141,324)(142,323)(143,322)(144,310)
(145,309)(146,319)(147,318)(148,317)(149,316)(150,315)(151,314)(152,313)
(153,312)(154,311)(155,299)(156,298)(157,308)(158,307)(159,306)(160,305)
(161,304)(162,303)(163,302)(164,301)(165,300);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(330)!( 56,111)( 57,112)( 58,113)( 59,114)( 60,115)( 61,116)( 62,117)
( 63,118)( 64,119)( 65,120)( 66,121)( 67,122)( 68,123)( 69,124)( 70,125)
( 71,126)( 72,127)( 73,128)( 74,129)( 75,130)( 76,131)( 77,132)( 78,133)
( 79,134)( 80,135)( 81,136)( 82,137)( 83,138)( 84,139)( 85,140)( 86,141)
( 87,142)( 88,143)( 89,144)( 90,145)( 91,146)( 92,147)( 93,148)( 94,149)
( 95,150)( 96,151)( 97,152)( 98,153)( 99,154)(100,155)(101,156)(102,157)
(103,158)(104,159)(105,160)(106,161)(107,162)(108,163)(109,164)(110,165)
(221,276)(222,277)(223,278)(224,279)(225,280)(226,281)(227,282)(228,283)
(229,284)(230,285)(231,286)(232,287)(233,288)(234,289)(235,290)(236,291)
(237,292)(238,293)(239,294)(240,295)(241,296)(242,297)(243,298)(244,299)
(245,300)(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)
(253,308)(254,309)(255,310)(256,311)(257,312)(258,313)(259,314)(260,315)
(261,316)(262,317)(263,318)(264,319)(265,320)(266,321)(267,322)(268,323)
(269,324)(270,325)(271,326)(272,327)(273,328)(274,329)(275,330);
s1 := Sym(330)!( 1, 56)( 2, 66)( 3, 65)( 4, 64)( 5, 63)( 6, 62)( 7, 61)
( 8, 60)( 9, 59)( 10, 58)( 11, 57)( 12,100)( 13,110)( 14,109)( 15,108)
( 16,107)( 17,106)( 18,105)( 19,104)( 20,103)( 21,102)( 22,101)( 23, 89)
( 24, 99)( 25, 98)( 26, 97)( 27, 96)( 28, 95)( 29, 94)( 30, 93)( 31, 92)
( 32, 91)( 33, 90)( 34, 78)( 35, 88)( 36, 87)( 37, 86)( 38, 85)( 39, 84)
( 40, 83)( 41, 82)( 42, 81)( 43, 80)( 44, 79)( 45, 67)( 46, 77)( 47, 76)
( 48, 75)( 49, 74)( 50, 73)( 51, 72)( 52, 71)( 53, 70)( 54, 69)( 55, 68)
(112,121)(113,120)(114,119)(115,118)(116,117)(122,155)(123,165)(124,164)
(125,163)(126,162)(127,161)(128,160)(129,159)(130,158)(131,157)(132,156)
(133,144)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)
(141,147)(142,146)(143,145)(166,221)(167,231)(168,230)(169,229)(170,228)
(171,227)(172,226)(173,225)(174,224)(175,223)(176,222)(177,265)(178,275)
(179,274)(180,273)(181,272)(182,271)(183,270)(184,269)(185,268)(186,267)
(187,266)(188,254)(189,264)(190,263)(191,262)(192,261)(193,260)(194,259)
(195,258)(196,257)(197,256)(198,255)(199,243)(200,253)(201,252)(202,251)
(203,250)(204,249)(205,248)(206,247)(207,246)(208,245)(209,244)(210,232)
(211,242)(212,241)(213,240)(214,239)(215,238)(216,237)(217,236)(218,235)
(219,234)(220,233)(277,286)(278,285)(279,284)(280,283)(281,282)(287,320)
(288,330)(289,329)(290,328)(291,327)(292,326)(293,325)(294,324)(295,323)
(296,322)(297,321)(298,309)(299,319)(300,318)(301,317)(302,316)(303,315)
(304,314)(305,313)(306,312)(307,311)(308,310);
s2 := Sym(330)!( 1,178)( 2,177)( 3,187)( 4,186)( 5,185)( 6,184)( 7,183)
( 8,182)( 9,181)( 10,180)( 11,179)( 12,167)( 13,166)( 14,176)( 15,175)
( 16,174)( 17,173)( 18,172)( 19,171)( 20,170)( 21,169)( 22,168)( 23,211)
( 24,210)( 25,220)( 26,219)( 27,218)( 28,217)( 29,216)( 30,215)( 31,214)
( 32,213)( 33,212)( 34,200)( 35,199)( 36,209)( 37,208)( 38,207)( 39,206)
( 40,205)( 41,204)( 42,203)( 43,202)( 44,201)( 45,189)( 46,188)( 47,198)
( 48,197)( 49,196)( 50,195)( 51,194)( 52,193)( 53,192)( 54,191)( 55,190)
( 56,233)( 57,232)( 58,242)( 59,241)( 60,240)( 61,239)( 62,238)( 63,237)
( 64,236)( 65,235)( 66,234)( 67,222)( 68,221)( 69,231)( 70,230)( 71,229)
( 72,228)( 73,227)( 74,226)( 75,225)( 76,224)( 77,223)( 78,266)( 79,265)
( 80,275)( 81,274)( 82,273)( 83,272)( 84,271)( 85,270)( 86,269)( 87,268)
( 88,267)( 89,255)( 90,254)( 91,264)( 92,263)( 93,262)( 94,261)( 95,260)
( 96,259)( 97,258)( 98,257)( 99,256)(100,244)(101,243)(102,253)(103,252)
(104,251)(105,250)(106,249)(107,248)(108,247)(109,246)(110,245)(111,288)
(112,287)(113,297)(114,296)(115,295)(116,294)(117,293)(118,292)(119,291)
(120,290)(121,289)(122,277)(123,276)(124,286)(125,285)(126,284)(127,283)
(128,282)(129,281)(130,280)(131,279)(132,278)(133,321)(134,320)(135,330)
(136,329)(137,328)(138,327)(139,326)(140,325)(141,324)(142,323)(143,322)
(144,310)(145,309)(146,319)(147,318)(148,317)(149,316)(150,315)(151,314)
(152,313)(153,312)(154,311)(155,299)(156,298)(157,308)(158,307)(159,306)
(160,305)(161,304)(162,303)(163,302)(164,301)(165,300);
poly := sub<Sym(330)|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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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