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Polytope of Type {40,18}

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