Polytope of Type {20,18}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,18}*1440
if this polytope has a name.
Group : SmallGroup(1440,4569)
Rank : 3
Schlafli Type : {20,18}
Number of vertices, edges, etc : 40, 360, 36
Order of s₀s₁s₂ : 90
Order of s₀s₁s₂s₁ : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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}*720b
   3-fold quotients : {20,6}*480c
   4-fold quotients : {10,18}*360
   5-fold quotients : {4,18}*288
   6-fold quotients : {20,6}*240b
   10-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c
   12-fold quotients : {10,6}*120
   15-fold quotients : {4,6}*96
   20-fold quotients : {4,9}*72, {2,18}*72
   30-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   36-fold quotients : {10,2}*40
   40-fold quotients : {2,9}*36
   60-fold quotients : {4,3}*24, {2,6}*24
   72-fold quotients : {5,2}*20
   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 := (  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 51)( 14, 52)
( 15, 49)( 16, 50)( 17, 55)( 18, 56)( 19, 53)( 20, 54)( 21, 59)( 22, 60)
( 23, 57)( 24, 58)( 25, 39)( 26, 40)( 27, 37)( 28, 38)( 29, 43)( 30, 44)
( 31, 41)( 32, 42)( 33, 47)( 34, 48)( 35, 45)( 36, 46)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73,111)( 74,112)( 75,109)( 76,110)
( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)( 84,118)
( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)( 91,101)( 92,102)
( 93,107)( 94,108)( 95,105)( 96,106)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,171)(134,172)(135,169)(136,170)(137,175)(138,176)
(139,173)(140,174)(141,179)(142,180)(143,177)(144,178)(145,159)(146,160)
(147,157)(148,158)(149,163)(150,164)(151,161)(152,162)(153,167)(154,168)
(155,165)(156,166)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,231)(194,232)(195,229)(196,230)(197,235)(198,236)(199,233)(200,234)
(201,239)(202,240)(203,237)(204,238)(205,219)(206,220)(207,217)(208,218)
(209,223)(210,224)(211,221)(212,222)(213,227)(214,228)(215,225)(216,226)
(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,291)(254,292)
(255,289)(256,290)(257,295)(258,296)(259,293)(260,294)(261,299)(262,300)
(263,297)(264,298)(265,279)(266,280)(267,277)(268,278)(269,283)(270,284)
(271,281)(272,282)(273,287)(274,288)(275,285)(276,286)(301,303)(302,304)
(305,307)(306,308)(309,311)(310,312)(313,351)(314,352)(315,349)(316,350)
(317,355)(318,356)(319,353)(320,354)(321,359)(322,360)(323,357)(324,358)
(325,339)(326,340)(327,337)(328,338)(329,343)(330,344)(331,341)(332,342)
(333,347)(334,348)(335,345)(336,346);;
s1 := (  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)(  8, 23)
(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 25, 49)( 26, 50)( 27, 52)( 28, 51)
( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)( 36, 55)
( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 61,141)( 62,142)( 63,144)
( 64,143)( 65,137)( 66,138)( 67,140)( 68,139)( 69,133)( 70,134)( 71,136)
( 72,135)( 73,129)( 74,130)( 75,132)( 76,131)( 77,125)( 78,126)( 79,128)
( 80,127)( 81,121)( 82,122)( 83,124)( 84,123)( 85,177)( 86,178)( 87,180)
( 88,179)( 89,173)( 90,174)( 91,176)( 92,175)( 93,169)( 94,170)( 95,172)
( 96,171)( 97,165)( 98,166)( 99,168)(100,167)(101,161)(102,162)(103,164)
(104,163)(105,157)(106,158)(107,160)(108,159)(109,153)(110,154)(111,156)
(112,155)(113,149)(114,150)(115,152)(116,151)(117,145)(118,146)(119,148)
(120,147)(181,193)(182,194)(183,196)(184,195)(185,201)(186,202)(187,204)
(188,203)(189,197)(190,198)(191,200)(192,199)(205,229)(206,230)(207,232)
(208,231)(209,237)(210,238)(211,240)(212,239)(213,233)(214,234)(215,236)
(216,235)(219,220)(221,225)(222,226)(223,228)(224,227)(241,321)(242,322)
(243,324)(244,323)(245,317)(246,318)(247,320)(248,319)(249,313)(250,314)
(251,316)(252,315)(253,309)(254,310)(255,312)(256,311)(257,305)(258,306)
(259,308)(260,307)(261,301)(262,302)(263,304)(264,303)(265,357)(266,358)
(267,360)(268,359)(269,353)(270,354)(271,356)(272,355)(273,349)(274,350)
(275,352)(276,351)(277,345)(278,346)(279,348)(280,347)(281,341)(282,342)
(283,344)(284,343)(285,337)(286,338)(287,340)(288,339)(289,333)(290,334)
(291,336)(292,335)(293,329)(294,330)(295,332)(296,331)(297,325)(298,326)
(299,328)(300,327);;
s2 := (  1,241)(  2,244)(  3,243)(  4,242)(  5,249)(  6,252)(  7,251)(  8,250)
(  9,245)( 10,248)( 11,247)( 12,246)( 13,253)( 14,256)( 15,255)( 16,254)
( 17,261)( 18,264)( 19,263)( 20,262)( 21,257)( 22,260)( 23,259)( 24,258)
( 25,265)( 26,268)( 27,267)( 28,266)( 29,273)( 30,276)( 31,275)( 32,274)
( 33,269)( 34,272)( 35,271)( 36,270)( 37,277)( 38,280)( 39,279)( 40,278)
( 41,285)( 42,288)( 43,287)( 44,286)( 45,281)( 46,284)( 47,283)( 48,282)
( 49,289)( 50,292)( 51,291)( 52,290)( 53,297)( 54,300)( 55,299)( 56,298)
( 57,293)( 58,296)( 59,295)( 60,294)( 61,181)( 62,184)( 63,183)( 64,182)
( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)( 72,186)
( 73,193)( 74,196)( 75,195)( 76,194)( 77,201)( 78,204)( 79,203)( 80,202)
( 81,197)( 82,200)( 83,199)( 84,198)( 85,205)( 86,208)( 87,207)( 88,206)
( 89,213)( 90,216)( 91,215)( 92,214)( 93,209)( 94,212)( 95,211)( 96,210)
( 97,217)( 98,220)( 99,219)(100,218)(101,225)(102,228)(103,227)(104,226)
(105,221)(106,224)(107,223)(108,222)(109,229)(110,232)(111,231)(112,230)
(113,237)(114,240)(115,239)(116,238)(117,233)(118,236)(119,235)(120,234)
(121,309)(122,312)(123,311)(124,310)(125,305)(126,308)(127,307)(128,306)
(129,301)(130,304)(131,303)(132,302)(133,321)(134,324)(135,323)(136,322)
(137,317)(138,320)(139,319)(140,318)(141,313)(142,316)(143,315)(144,314)
(145,333)(146,336)(147,335)(148,334)(149,329)(150,332)(151,331)(152,330)
(153,325)(154,328)(155,327)(156,326)(157,345)(158,348)(159,347)(160,346)
(161,341)(162,344)(163,343)(164,342)(165,337)(166,340)(167,339)(168,338)
(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)(176,354)
(177,349)(178,352)(179,351)(180,350);;
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(360)!(  1,  3)(  2,  4)(  5,  7)(  6,  8)(  9, 11)( 10, 12)( 13, 51)
( 14, 52)( 15, 49)( 16, 50)( 17, 55)( 18, 56)( 19, 53)( 20, 54)( 21, 59)
( 22, 60)( 23, 57)( 24, 58)( 25, 39)( 26, 40)( 27, 37)( 28, 38)( 29, 43)
( 30, 44)( 31, 41)( 32, 42)( 33, 47)( 34, 48)( 35, 45)( 36, 46)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73,111)( 74,112)( 75,109)
( 76,110)( 77,115)( 78,116)( 79,113)( 80,114)( 81,119)( 82,120)( 83,117)
( 84,118)( 85, 99)( 86,100)( 87, 97)( 88, 98)( 89,103)( 90,104)( 91,101)
( 92,102)( 93,107)( 94,108)( 95,105)( 96,106)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,171)(134,172)(135,169)(136,170)(137,175)
(138,176)(139,173)(140,174)(141,179)(142,180)(143,177)(144,178)(145,159)
(146,160)(147,157)(148,158)(149,163)(150,164)(151,161)(152,162)(153,167)
(154,168)(155,165)(156,166)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,231)(194,232)(195,229)(196,230)(197,235)(198,236)(199,233)
(200,234)(201,239)(202,240)(203,237)(204,238)(205,219)(206,220)(207,217)
(208,218)(209,223)(210,224)(211,221)(212,222)(213,227)(214,228)(215,225)
(216,226)(241,243)(242,244)(245,247)(246,248)(249,251)(250,252)(253,291)
(254,292)(255,289)(256,290)(257,295)(258,296)(259,293)(260,294)(261,299)
(262,300)(263,297)(264,298)(265,279)(266,280)(267,277)(268,278)(269,283)
(270,284)(271,281)(272,282)(273,287)(274,288)(275,285)(276,286)(301,303)
(302,304)(305,307)(306,308)(309,311)(310,312)(313,351)(314,352)(315,349)
(316,350)(317,355)(318,356)(319,353)(320,354)(321,359)(322,360)(323,357)
(324,358)(325,339)(326,340)(327,337)(328,338)(329,343)(330,344)(331,341)
(332,342)(333,347)(334,348)(335,345)(336,346);
s1 := Sym(360)!(  1, 13)(  2, 14)(  3, 16)(  4, 15)(  5, 21)(  6, 22)(  7, 24)
(  8, 23)(  9, 17)( 10, 18)( 11, 20)( 12, 19)( 25, 49)( 26, 50)( 27, 52)
( 28, 51)( 29, 57)( 30, 58)( 31, 60)( 32, 59)( 33, 53)( 34, 54)( 35, 56)
( 36, 55)( 39, 40)( 41, 45)( 42, 46)( 43, 48)( 44, 47)( 61,141)( 62,142)
( 63,144)( 64,143)( 65,137)( 66,138)( 67,140)( 68,139)( 69,133)( 70,134)
( 71,136)( 72,135)( 73,129)( 74,130)( 75,132)( 76,131)( 77,125)( 78,126)
( 79,128)( 80,127)( 81,121)( 82,122)( 83,124)( 84,123)( 85,177)( 86,178)
( 87,180)( 88,179)( 89,173)( 90,174)( 91,176)( 92,175)( 93,169)( 94,170)
( 95,172)( 96,171)( 97,165)( 98,166)( 99,168)(100,167)(101,161)(102,162)
(103,164)(104,163)(105,157)(106,158)(107,160)(108,159)(109,153)(110,154)
(111,156)(112,155)(113,149)(114,150)(115,152)(116,151)(117,145)(118,146)
(119,148)(120,147)(181,193)(182,194)(183,196)(184,195)(185,201)(186,202)
(187,204)(188,203)(189,197)(190,198)(191,200)(192,199)(205,229)(206,230)
(207,232)(208,231)(209,237)(210,238)(211,240)(212,239)(213,233)(214,234)
(215,236)(216,235)(219,220)(221,225)(222,226)(223,228)(224,227)(241,321)
(242,322)(243,324)(244,323)(245,317)(246,318)(247,320)(248,319)(249,313)
(250,314)(251,316)(252,315)(253,309)(254,310)(255,312)(256,311)(257,305)
(258,306)(259,308)(260,307)(261,301)(262,302)(263,304)(264,303)(265,357)
(266,358)(267,360)(268,359)(269,353)(270,354)(271,356)(272,355)(273,349)
(274,350)(275,352)(276,351)(277,345)(278,346)(279,348)(280,347)(281,341)
(282,342)(283,344)(284,343)(285,337)(286,338)(287,340)(288,339)(289,333)
(290,334)(291,336)(292,335)(293,329)(294,330)(295,332)(296,331)(297,325)
(298,326)(299,328)(300,327);
s2 := Sym(360)!(  1,241)(  2,244)(  3,243)(  4,242)(  5,249)(  6,252)(  7,251)
(  8,250)(  9,245)( 10,248)( 11,247)( 12,246)( 13,253)( 14,256)( 15,255)
( 16,254)( 17,261)( 18,264)( 19,263)( 20,262)( 21,257)( 22,260)( 23,259)
( 24,258)( 25,265)( 26,268)( 27,267)( 28,266)( 29,273)( 30,276)( 31,275)
( 32,274)( 33,269)( 34,272)( 35,271)( 36,270)( 37,277)( 38,280)( 39,279)
( 40,278)( 41,285)( 42,288)( 43,287)( 44,286)( 45,281)( 46,284)( 47,283)
( 48,282)( 49,289)( 50,292)( 51,291)( 52,290)( 53,297)( 54,300)( 55,299)
( 56,298)( 57,293)( 58,296)( 59,295)( 60,294)( 61,181)( 62,184)( 63,183)
( 64,182)( 65,189)( 66,192)( 67,191)( 68,190)( 69,185)( 70,188)( 71,187)
( 72,186)( 73,193)( 74,196)( 75,195)( 76,194)( 77,201)( 78,204)( 79,203)
( 80,202)( 81,197)( 82,200)( 83,199)( 84,198)( 85,205)( 86,208)( 87,207)
( 88,206)( 89,213)( 90,216)( 91,215)( 92,214)( 93,209)( 94,212)( 95,211)
( 96,210)( 97,217)( 98,220)( 99,219)(100,218)(101,225)(102,228)(103,227)
(104,226)(105,221)(106,224)(107,223)(108,222)(109,229)(110,232)(111,231)
(112,230)(113,237)(114,240)(115,239)(116,238)(117,233)(118,236)(119,235)
(120,234)(121,309)(122,312)(123,311)(124,310)(125,305)(126,308)(127,307)
(128,306)(129,301)(130,304)(131,303)(132,302)(133,321)(134,324)(135,323)
(136,322)(137,317)(138,320)(139,319)(140,318)(141,313)(142,316)(143,315)
(144,314)(145,333)(146,336)(147,335)(148,334)(149,329)(150,332)(151,331)
(152,330)(153,325)(154,328)(155,327)(156,326)(157,345)(158,348)(159,347)
(160,346)(161,341)(162,344)(163,343)(164,342)(165,337)(166,340)(167,339)
(168,338)(169,357)(170,360)(171,359)(172,358)(173,353)(174,356)(175,355)
(176,354)(177,349)(178,352)(179,351)(180,350);
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*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s1*s2*s1*s2*s1*s0*s1, 
s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*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 >;

References : None.
to this polytope