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}*640b
if this polytope has a name.
Group : SmallGroup(640,21459)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 16, 160, 80
Order of s0s1s2 : 20
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Non-Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {4,20,2} of size 1280
Vertex Figure Of :
   {2,4,20} of size 1280
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,10}*320a
   4-fold quotients : {4,5}*160
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,20}*1280i, {8,20}*1280j, {4,40}*1280e, {4,40}*1280f, {4,20}*1280c
   3-fold covers : {4,60}*1920f
Irregular Quotients (of which this is a minimal cover):
   P/N, where N=<s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2> of order 2.
      40 facets:
         40 of {4}*8
      8 vertex figures:
         8 of {20}*40
   P/N, where N=<s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      40 facets:
         40 of {4}*8
      8 vertex figures:
         8 of {20}*40
   P/N, where N=<s0*s2*s1*s0*s1*s2> of order 2.
      48 facets:
         32 of {4}*8
         16 of {2}*4
      8 vertex figures:
         8 of {20}*40
   P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1> of order 4.
      24 facets:
         16 of {4}*8
         8 of {2}*4
      4 vertex figures:
         4 of {20}*40
   P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s0*s2*s1*s0*s1*s2*s1> of order 4.
      28 facets:
         12 of {4}*8
         16 of {2}*4
      4 vertex figures:
         4 of {20}*40
   P/N, where N=<s0*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      24 facets:
         16 of {4}*8
         8 of {2}*4
      4 vertex figures:
         4 of {20}*40

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