Polytope of Type {20,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {20,4}*1440
if this polytope has a name.
Group : SmallGroup(1440,4764)
Rank : 3
Schlafli Type : {20,4}
Number of vertices, edges, etc : 180, 360, 36
Order of s0s1s2 : 60
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Skewing Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {20,4}*720
   5-fold quotients : {4,4}*288
   9-fold quotients : {20,4}*160
   10-fold quotients : {4,4}*144
   18-fold quotients : {20,2}*80, {10,4}*80
   20-fold quotients : {4,4}*72
   36-fold quotients : {10,2}*40
   45-fold quotients : {4,4}*32
   72-fold quotients : {5,2}*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=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      20 facets:
         4 of {10}*20
         16 of {20}*40
      90 vertex figures:
         90 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 2.
      18 facets:
         18 of {20}*40
      90 vertex figures:
         90 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      18 facets:
         18 of {20}*40
      90 vertex figures:
         90 of {4}*8
   P/N, where N=<s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      18 facets:
         18 of {20}*40
      90 vertex figures:
         90 of {4}*8
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 2.
      18 facets:
         18 of {20}*40
      100 vertex figures:
         80 of {4}*8
         20 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 3.
      12 facets:
         12 of {20}*40
      60 vertex figures:
         60 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 3.
      12 facets:
         12 of {20}*40
      60 vertex figures:
         60 of {4}*8
   P/N, where N=<s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
      9 facets:
         9 of {20}*40
      50 vertex figures:
         40 of {4}*8
         10 of {2}*4
   P/N, where N=<s0*s2*s1*s0*s2*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 4.
      10 facets:
         2 of {10}*20
         8 of {20}*40
      45 vertex figures:
         45 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
      9 facets:
         9 of {20}*40
      45 vertex figures:
         45 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s2*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 6.
      6 facets:
         6 of {20}*40
      40 vertex figures:
         20 of {4}*8
         20 of {2}*4
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s2> of order 6.
      6 facets:
         6 of {20}*40
      30 vertex figures:
         30 of {4}*8
   P/N, where N=<s0*s2*s1*s0*s1*s2*s1*s0*s2*s1> of order 6.
      6 facets:
         6 of {20}*40
      30 vertex figures:
         30 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 6.
      6 facets:
         6 of {20}*40
      30 vertex figures:
         30 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 6.
      6 facets:
         6 of {20}*40
      30 vertex figures:
         30 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 6.
      8 facets:
         4 of {10}*20
         4 of {20}*40
      30 vertex figures:
         30 of {4}*8
   P/N, where N=<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> of order 6.
      8 facets:
         4 of {10}*20
         4 of {20}*40
      30 vertex figures:
         30 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0> of order 6.
      6 facets:
         6 of {20}*40
      40 vertex figures:
         20 of {2}*4
         20 of {4}*8

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

Twisty Puzzle