Polytope of Type {4,20}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,20}*1440
if this polytope has a name.
Group : SmallGroup(1440,4764)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 36, 360, 180
Order of s0s1s2 : 60
Order of s0s1s2s1 : 6
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 : {4,20}*720
   5-fold quotients : {4,4}*288
   9-fold quotients : {4,20}*160
   10-fold quotients : {4,4}*144
   18-fold quotients : {2,20}*80, {4,10}*80
   20-fold quotients : {4,4}*72
   36-fold quotients : {2,10}*40
   45-fold quotients : {4,4}*32
   72-fold quotients : {2,5}*20
   90-fold quotients : {2,4}*16, {4,2}*16
   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=<s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      90 facets:
         90 of {4}*8
      20 vertex figures:
         16 of {20}*40
         4 of {10}*20
   P/N, where N=<s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      90 facets:
         90 of {4}*8
      18 vertex figures:
         18 of {20}*40
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      90 facets:
         90 of {4}*8
      18 vertex figures:
         18 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 2.
      90 facets:
         90 of {4}*8
      18 vertex figures:
         18 of {20}*40
   P/N, where N=<s0*s1*s0*s1> of order 2.
      100 facets:
         20 of {2}*4
         80 of {4}*8
      18 vertex figures:
         18 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      60 facets:
         60 of {4}*8
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      60 facets:
         60 of {4}*8
      12 vertex figures:
         12 of {20}*40
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      50 facets:
         10 of {2}*4
         40 of {4}*8
      9 vertex figures:
         9 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
      45 facets:
         45 of {4}*8
      10 vertex figures:
         8 of {20}*40
         2 of {10}*20
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2, s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1> of order 4.
      45 facets:
         45 of {4}*8
      9 vertex figures:
         9 of {20}*40
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      40 facets:
         20 of {2}*4
         20 of {4}*8
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 6.
      30 facets:
         30 of {4}*8
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s2> of order 6.
      30 facets:
         30 of {4}*8
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 6.
      30 facets:
         30 of {4}*8
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      30 facets:
         30 of {4}*8
      6 vertex figures:
         6 of {20}*40
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      30 facets:
         30 of {4}*8
      8 vertex figures:
         4 of {10}*20
         4 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 6.
      30 facets:
         30 of {4}*8
      8 vertex figures:
         4 of {20}*40
         4 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 6.
      40 facets:
         20 of {2}*4
         20 of {4}*8
      6 vertex figures:
         6 of {20}*40

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

Twisty Puzzle