Polytope of Type {18,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {18,20}*1440
if this polytope has a name.
Group : SmallGroup(1440,4569)
Rank : 3
Schlafli Type : {18,20}
Number of vertices, edges, etc : 36, 360, 40
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 : {18,20}*720b
   3-fold quotients : {6,20}*480c
   4-fold quotients : {18,10}*360
   5-fold quotients : {18,4}*288
   6-fold quotients : {6,20}*240b
   10-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c
   12-fold quotients : {6,10}*120
   15-fold quotients : {6,4}*96
   20-fold quotients : {9,4}*72, {18,2}*72
   30-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
   36-fold quotients : {2,10}*40
   40-fold quotients : {9,2}*36
   60-fold quotients : {3,4}*24, {6,2}*24
   72-fold quotients : {2,5}*20
   120-fold quotients : {3,2}*12
   180-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      20 facets:
         20 of {18}*36
      24 vertex figures:
         12 of {20}*40
         12 of {10}*20
   P/N, where N=<s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2> of order 2.
      20 facets:
         20 of {18}*36
      18 vertex figures:
         18 of {20}*40

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

Twisty Puzzle