Polytope of Type {6,120}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,120}*1440b
Also Known As : {6,120|2}. if this polytope has another name.
Group : SmallGroup(1440,3583)
Rank : 3
Schlafli Type : {6,120}
Number of vertices, edges, etc : 6, 360, 120
Order of s0s1s2 : 120
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 : {6,60}*720b
3-fold quotients : {6,40}*480, {2,120}*480
4-fold quotients : {6,30}*360b
5-fold quotients : {6,24}*288a
6-fold quotients : {6,20}*240a, {2,60}*240
9-fold quotients : {2,40}*160
10-fold quotients : {6,12}*144a
12-fold quotients : {6,10}*120, {2,30}*120
15-fold quotients : {2,24}*96, {6,8}*96
18-fold quotients : {2,20}*80
20-fold quotients : {6,6}*72a
24-fold quotients : {2,15}*60
30-fold quotients : {2,12}*48, {6,4}*48a
36-fold quotients : {2,10}*40
45-fold quotients : {2,8}*32
60-fold quotients : {2,6}*24, {6,2}*24
72-fold quotients : {2,5}*20
90-fold quotients : {2,4}*16
120-fold quotients : {2,3}*12, {3,2}*12
180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)( 23, 38)
( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)( 61, 76)
( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)( 69, 84)
( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)(106,121)(107,122)
(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)(115,130)
(116,131)(117,132)(118,133)(119,134)(120,135)(151,166)(152,167)(153,168)
(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)(161,176)
(162,177)(163,178)(164,179)(165,180)(196,211)(197,212)(198,213)(199,214)
(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)(207,222)
(208,223)(209,224)(210,225)(241,256)(242,257)(243,258)(244,259)(245,260)
(246,261)(247,262)(248,263)(249,264)(250,265)(251,266)(252,267)(253,268)
(254,269)(255,270)(286,301)(287,302)(288,303)(289,304)(290,305)(291,306)
(292,307)(293,308)(294,309)(295,310)(296,311)(297,312)(298,313)(299,314)
(300,315)(331,346)(332,347)(333,348)(334,349)(335,350)(336,351)(337,352)
(338,353)(339,354)(340,355)(341,356)(342,357)(343,358)(344,359)(345,360);;
s1 := ( 1, 16)( 2, 20)( 3, 19)( 4, 18)( 5, 17)( 6, 26)( 7, 30)( 8, 29)
( 9, 28)( 10, 27)( 11, 21)( 12, 25)( 13, 24)( 14, 23)( 15, 22)( 32, 35)
( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 61)( 47, 65)
( 48, 64)( 49, 63)( 50, 62)( 51, 71)( 52, 75)( 53, 74)( 54, 73)( 55, 72)
( 56, 66)( 57, 70)( 58, 69)( 59, 68)( 60, 67)( 77, 80)( 78, 79)( 81, 86)
( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 91,151)( 92,155)( 93,154)( 94,153)
( 95,152)( 96,161)( 97,165)( 98,164)( 99,163)(100,162)(101,156)(102,160)
(103,159)(104,158)(105,157)(106,136)(107,140)(108,139)(109,138)(110,137)
(111,146)(112,150)(113,149)(114,148)(115,147)(116,141)(117,145)(118,144)
(119,143)(120,142)(121,166)(122,170)(123,169)(124,168)(125,167)(126,176)
(127,180)(128,179)(129,178)(130,177)(131,171)(132,175)(133,174)(134,173)
(135,172)(181,286)(182,290)(183,289)(184,288)(185,287)(186,296)(187,300)
(188,299)(189,298)(190,297)(191,291)(192,295)(193,294)(194,293)(195,292)
(196,271)(197,275)(198,274)(199,273)(200,272)(201,281)(202,285)(203,284)
(204,283)(205,282)(206,276)(207,280)(208,279)(209,278)(210,277)(211,301)
(212,305)(213,304)(214,303)(215,302)(216,311)(217,315)(218,314)(219,313)
(220,312)(221,306)(222,310)(223,309)(224,308)(225,307)(226,331)(227,335)
(228,334)(229,333)(230,332)(231,341)(232,345)(233,344)(234,343)(235,342)
(236,336)(237,340)(238,339)(239,338)(240,337)(241,316)(242,320)(243,319)
(244,318)(245,317)(246,326)(247,330)(248,329)(249,328)(250,327)(251,321)
(252,325)(253,324)(254,323)(255,322)(256,346)(257,350)(258,349)(259,348)
(260,347)(261,356)(262,360)(263,359)(264,358)(265,357)(266,351)(267,355)
(268,354)(269,353)(270,352);;
s2 := ( 1,187)( 2,186)( 3,190)( 4,189)( 5,188)( 6,182)( 7,181)( 8,185)
( 9,184)( 10,183)( 11,192)( 12,191)( 13,195)( 14,194)( 15,193)( 16,202)
( 17,201)( 18,205)( 19,204)( 20,203)( 21,197)( 22,196)( 23,200)( 24,199)
( 25,198)( 26,207)( 27,206)( 28,210)( 29,209)( 30,208)( 31,217)( 32,216)
( 33,220)( 34,219)( 35,218)( 36,212)( 37,211)( 38,215)( 39,214)( 40,213)
( 41,222)( 42,221)( 43,225)( 44,224)( 45,223)( 46,232)( 47,231)( 48,235)
( 49,234)( 50,233)( 51,227)( 52,226)( 53,230)( 54,229)( 55,228)( 56,237)
( 57,236)( 58,240)( 59,239)( 60,238)( 61,247)( 62,246)( 63,250)( 64,249)
( 65,248)( 66,242)( 67,241)( 68,245)( 69,244)( 70,243)( 71,252)( 72,251)
( 73,255)( 74,254)( 75,253)( 76,262)( 77,261)( 78,265)( 79,264)( 80,263)
( 81,257)( 82,256)( 83,260)( 84,259)( 85,258)( 86,267)( 87,266)( 88,270)
( 89,269)( 90,268)( 91,322)( 92,321)( 93,325)( 94,324)( 95,323)( 96,317)
( 97,316)( 98,320)( 99,319)(100,318)(101,327)(102,326)(103,330)(104,329)
(105,328)(106,337)(107,336)(108,340)(109,339)(110,338)(111,332)(112,331)
(113,335)(114,334)(115,333)(116,342)(117,341)(118,345)(119,344)(120,343)
(121,352)(122,351)(123,355)(124,354)(125,353)(126,347)(127,346)(128,350)
(129,349)(130,348)(131,357)(132,356)(133,360)(134,359)(135,358)(136,277)
(137,276)(138,280)(139,279)(140,278)(141,272)(142,271)(143,275)(144,274)
(145,273)(146,282)(147,281)(148,285)(149,284)(150,283)(151,292)(152,291)
(153,295)(154,294)(155,293)(156,287)(157,286)(158,290)(159,289)(160,288)
(161,297)(162,296)(163,300)(164,299)(165,298)(166,307)(167,306)(168,310)
(169,309)(170,308)(171,302)(172,301)(173,305)(174,304)(175,303)(176,312)
(177,311)(178,315)(179,314)(180,313);;
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*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(360)!( 16, 31)( 17, 32)( 18, 33)( 19, 34)( 20, 35)( 21, 36)( 22, 37)
( 23, 38)( 24, 39)( 25, 40)( 26, 41)( 27, 42)( 28, 43)( 29, 44)( 30, 45)
( 61, 76)( 62, 77)( 63, 78)( 64, 79)( 65, 80)( 66, 81)( 67, 82)( 68, 83)
( 69, 84)( 70, 85)( 71, 86)( 72, 87)( 73, 88)( 74, 89)( 75, 90)(106,121)
(107,122)(108,123)(109,124)(110,125)(111,126)(112,127)(113,128)(114,129)
(115,130)(116,131)(117,132)(118,133)(119,134)(120,135)(151,166)(152,167)
(153,168)(154,169)(155,170)(156,171)(157,172)(158,173)(159,174)(160,175)
(161,176)(162,177)(163,178)(164,179)(165,180)(196,211)(197,212)(198,213)
(199,214)(200,215)(201,216)(202,217)(203,218)(204,219)(205,220)(206,221)
(207,222)(208,223)(209,224)(210,225)(241,256)(242,257)(243,258)(244,259)
(245,260)(246,261)(247,262)(248,263)(249,264)(250,265)(251,266)(252,267)
(253,268)(254,269)(255,270)(286,301)(287,302)(288,303)(289,304)(290,305)
(291,306)(292,307)(293,308)(294,309)(295,310)(296,311)(297,312)(298,313)
(299,314)(300,315)(331,346)(332,347)(333,348)(334,349)(335,350)(336,351)
(337,352)(338,353)(339,354)(340,355)(341,356)(342,357)(343,358)(344,359)
(345,360);
s1 := Sym(360)!( 1, 16)( 2, 20)( 3, 19)( 4, 18)( 5, 17)( 6, 26)( 7, 30)
( 8, 29)( 9, 28)( 10, 27)( 11, 21)( 12, 25)( 13, 24)( 14, 23)( 15, 22)
( 32, 35)( 33, 34)( 36, 41)( 37, 45)( 38, 44)( 39, 43)( 40, 42)( 46, 61)
( 47, 65)( 48, 64)( 49, 63)( 50, 62)( 51, 71)( 52, 75)( 53, 74)( 54, 73)
( 55, 72)( 56, 66)( 57, 70)( 58, 69)( 59, 68)( 60, 67)( 77, 80)( 78, 79)
( 81, 86)( 82, 90)( 83, 89)( 84, 88)( 85, 87)( 91,151)( 92,155)( 93,154)
( 94,153)( 95,152)( 96,161)( 97,165)( 98,164)( 99,163)(100,162)(101,156)
(102,160)(103,159)(104,158)(105,157)(106,136)(107,140)(108,139)(109,138)
(110,137)(111,146)(112,150)(113,149)(114,148)(115,147)(116,141)(117,145)
(118,144)(119,143)(120,142)(121,166)(122,170)(123,169)(124,168)(125,167)
(126,176)(127,180)(128,179)(129,178)(130,177)(131,171)(132,175)(133,174)
(134,173)(135,172)(181,286)(182,290)(183,289)(184,288)(185,287)(186,296)
(187,300)(188,299)(189,298)(190,297)(191,291)(192,295)(193,294)(194,293)
(195,292)(196,271)(197,275)(198,274)(199,273)(200,272)(201,281)(202,285)
(203,284)(204,283)(205,282)(206,276)(207,280)(208,279)(209,278)(210,277)
(211,301)(212,305)(213,304)(214,303)(215,302)(216,311)(217,315)(218,314)
(219,313)(220,312)(221,306)(222,310)(223,309)(224,308)(225,307)(226,331)
(227,335)(228,334)(229,333)(230,332)(231,341)(232,345)(233,344)(234,343)
(235,342)(236,336)(237,340)(238,339)(239,338)(240,337)(241,316)(242,320)
(243,319)(244,318)(245,317)(246,326)(247,330)(248,329)(249,328)(250,327)
(251,321)(252,325)(253,324)(254,323)(255,322)(256,346)(257,350)(258,349)
(259,348)(260,347)(261,356)(262,360)(263,359)(264,358)(265,357)(266,351)
(267,355)(268,354)(269,353)(270,352);
s2 := Sym(360)!( 1,187)( 2,186)( 3,190)( 4,189)( 5,188)( 6,182)( 7,181)
( 8,185)( 9,184)( 10,183)( 11,192)( 12,191)( 13,195)( 14,194)( 15,193)
( 16,202)( 17,201)( 18,205)( 19,204)( 20,203)( 21,197)( 22,196)( 23,200)
( 24,199)( 25,198)( 26,207)( 27,206)( 28,210)( 29,209)( 30,208)( 31,217)
( 32,216)( 33,220)( 34,219)( 35,218)( 36,212)( 37,211)( 38,215)( 39,214)
( 40,213)( 41,222)( 42,221)( 43,225)( 44,224)( 45,223)( 46,232)( 47,231)
( 48,235)( 49,234)( 50,233)( 51,227)( 52,226)( 53,230)( 54,229)( 55,228)
( 56,237)( 57,236)( 58,240)( 59,239)( 60,238)( 61,247)( 62,246)( 63,250)
( 64,249)( 65,248)( 66,242)( 67,241)( 68,245)( 69,244)( 70,243)( 71,252)
( 72,251)( 73,255)( 74,254)( 75,253)( 76,262)( 77,261)( 78,265)( 79,264)
( 80,263)( 81,257)( 82,256)( 83,260)( 84,259)( 85,258)( 86,267)( 87,266)
( 88,270)( 89,269)( 90,268)( 91,322)( 92,321)( 93,325)( 94,324)( 95,323)
( 96,317)( 97,316)( 98,320)( 99,319)(100,318)(101,327)(102,326)(103,330)
(104,329)(105,328)(106,337)(107,336)(108,340)(109,339)(110,338)(111,332)
(112,331)(113,335)(114,334)(115,333)(116,342)(117,341)(118,345)(119,344)
(120,343)(121,352)(122,351)(123,355)(124,354)(125,353)(126,347)(127,346)
(128,350)(129,349)(130,348)(131,357)(132,356)(133,360)(134,359)(135,358)
(136,277)(137,276)(138,280)(139,279)(140,278)(141,272)(142,271)(143,275)
(144,274)(145,273)(146,282)(147,281)(148,285)(149,284)(150,283)(151,292)
(152,291)(153,295)(154,294)(155,293)(156,287)(157,286)(158,290)(159,289)
(160,288)(161,297)(162,296)(163,300)(164,299)(165,298)(166,307)(167,306)
(168,310)(169,309)(170,308)(171,302)(172,301)(173,305)(174,304)(175,303)
(176,312)(177,311)(178,315)(179,314)(180,313);
poly := sub<Sym(360)|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*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