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