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}*2000a
if this polytope has a name.
Group : SmallGroup(2000,482)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 50, 500, 250
Order of s0s1s2 : 10
Order of s0s1s2s1 : 10
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,20}*1000
   5-fold quotients : {4,4}*400
   10-fold quotients : {4,4}*200
   250-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*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      125 facets:
         125 of {4}*8
      25 vertex figures:
         25 of {20}*40
   P/N, where N=<s0*s1*s0*s1> of order 2.
      130 facets:
         10 of {2}*4
         120 of {4}*8
      26 vertex figures:
         24 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      65 facets:
         5 of {2}*4
         60 of {4}*8
      13 vertex figures:
         12 of {20}*40
         1 of {10}*20
   P/N, where N=<s0*s1*s0*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1> of order 5.
      50 facets:
         50 of {4}*8
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2> of order 5.
      50 facets:
         50 of {4}*8
      10 vertex figures:
         10 of {20}*40
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1> of order 10.
      30 facets:
         10 of {2}*4
         20 of {4}*8
      6 vertex figures:
         4 of {20}*40
         2 of {10}*20
   P/N, where N=<s0*s1*s0*s1, s0*s1*s2*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2> of order 10.
      30 facets:
         10 of {2}*4
         20 of {4}*8
      6 vertex figures:
         4 of {20}*40
         2 of {10}*20

Permutation Representation (GAP) : 
s0 := (  2,  5)(  3,  4)(  6, 59)(  7, 58)(  8, 57)(  9, 56)( 10, 60)( 11,114)( 12,113)( 13,112)( 14,111)( 15,115)( 16, 41)( 17, 45)( 18, 44)( 19, 43)( 20, 42)( 21,100)( 22, 99)( 23, 98)( 24, 97)( 25, 96)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31, 34)( 32, 33)( 36, 89)( 37, 88)( 38, 87)( 39, 86)( 40, 90)( 46, 75)( 47, 74)( 48, 73)( 49, 72)( 50, 71)( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 61, 64)( 62, 63)( 66,116)( 67,120)( 68,119)( 69,118)( 70,117)( 81,109)( 82,108)( 83,107)( 84,106)( 85,110)( 92, 95)( 93, 94)(121,125)(122,124)(127,130)(128,129)(131,184)(132,183)(133,182)(134,181)(135,185)(136,239)(137,238)(138,237)(139,236)(140,240)(141,166)(142,170)(143,169)(144,168)(145,167)(146,225)(147,224)(148,223)(149,222)(150,221)(151,226)(152,230)(153,229)(154,228)(155,227)(156,159)(157,158)(161,214)(162,213)(163,212)(164,211)(165,215)(171,200)(172,199)(173,198)(174,197)(175,196)(176,201)(177,205)(178,204)(179,203)(180,202)(186,189)(187,188)(191,241)(192,245)(193,244)(194,243)(195,242)(206,234)(207,233)(208,232)(209,231)(210,235)(217,220)(218,219)(246,250)(247,249);;
s1 := (  2,  5)(  3,  4)(  6, 41)(  7, 45)(  8, 44)(  9, 43)( 10, 42)( 11, 59)( 12, 58)( 13, 57)( 14, 56)( 15, 60)( 16,100)( 17, 99)( 18, 98)( 19, 97)( 20, 96)( 21,114)( 22,113)( 23,112)( 24,111)( 25,115)( 26, 61)( 27, 65)( 28, 64)( 29, 63)( 30, 62)( 31, 78)( 32, 77)( 33, 76)( 34, 80)( 35, 79)( 36,118)( 37,117)( 38,116)( 39,120)( 40,119)( 46, 47)( 48, 50)( 51,125)( 52,124)( 53,123)( 54,122)( 55,121)( 67, 70)( 68, 69)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 85)( 86, 88)( 89, 90)( 91,105)( 92,104)( 93,103)( 94,102)( 95,101)(106,108)(109,110)(127,130)(128,129)(131,166)(132,170)(133,169)(134,168)(135,167)(136,184)(137,183)(138,182)(139,181)(140,185)(141,225)(142,224)(143,223)(144,222)(145,221)(146,239)(147,238)(148,237)(149,236)(150,240)(151,186)(152,190)(153,189)(154,188)(155,187)(156,203)(157,202)(158,201)(159,205)(160,204)(161,243)(162,242)(163,241)(164,245)(165,244)(171,172)(173,175)(176,250)(177,249)(178,248)(179,247)(180,246)(192,195)(193,194)(196,209)(197,208)(198,207)(199,206)(200,210)(211,213)(214,215)(216,230)(217,229)(218,228)(219,227)(220,226)(231,233)(234,235);;
s2 := (  1,216)(  2,220)(  3,219)(  4,218)(  5,217)(  6,162)(  7,161)(  8,165)(  9,164)( 10,163)( 11,235)( 12,234)( 13,233)( 14,232)( 15,231)( 16,180)( 17,179)( 18,178)( 19,177)( 20,176)( 21,147)( 22,146)( 23,150)( 24,149)( 25,148)( 26,243)( 27,242)( 28,241)( 29,245)( 30,244)( 31,189)( 32,188)( 33,187)( 34,186)( 35,190)( 36,132)( 37,131)( 38,135)( 39,134)( 40,133)( 41,202)( 42,201)( 43,205)( 44,204)( 45,203)( 46,174)( 47,173)( 48,172)( 49,171)( 50,175)( 51,145)( 52,144)( 53,143)( 54,142)( 55,141)( 56,211)( 57,215)( 58,214)( 59,213)( 60,212)( 61,159)( 62,158)( 63,157)( 64,156)( 65,160)( 66,229)( 67,228)( 68,227)( 69,226)( 70,230)( 71,196)( 72,200)( 73,199)( 74,198)( 75,197)( 76,167)( 77,166)( 78,170)( 79,169)( 80,168)( 81,238)( 82,237)( 83,236)( 84,240)( 85,239)( 86,181)( 87,185)( 88,184)( 89,183)( 90,182)( 91,126)( 92,130)( 93,129)( 94,128)( 95,127)( 96,223)( 97,222)( 98,221)( 99,225)(100,224)(101,194)(102,193)(103,192)(104,191)(105,195)(106,140)(107,139)(108,138)(109,137)(110,136)(111,208)(112,207)(113,206)(114,210)(115,209)(116,153)(117,152)(118,151)(119,155)(120,154)(121,250)(122,249)(123,248)(124,247)(125,246);;
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, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(250)!(  2,  5)(  3,  4)(  6, 59)(  7, 58)(  8, 57)(  9, 56)( 10, 60)( 11,114)( 12,113)( 13,112)( 14,111)( 15,115)( 16, 41)( 17, 45)( 18, 44)( 19, 43)( 20, 42)( 21,100)( 22, 99)( 23, 98)( 24, 97)( 25, 96)( 26,101)( 27,105)( 28,104)( 29,103)( 30,102)( 31, 34)( 32, 33)( 36, 89)( 37, 88)( 38, 87)( 39, 86)( 40, 90)( 46, 75)( 47, 74)( 48, 73)( 49, 72)( 50, 71)( 51, 76)( 52, 80)( 53, 79)( 54, 78)( 55, 77)( 61, 64)( 62, 63)( 66,116)( 67,120)( 68,119)( 69,118)( 70,117)( 81,109)( 82,108)( 83,107)( 84,106)( 85,110)( 92, 95)( 93, 94)(121,125)(122,124)(127,130)(128,129)(131,184)(132,183)(133,182)(134,181)(135,185)(136,239)(137,238)(138,237)(139,236)(140,240)(141,166)(142,170)(143,169)(144,168)(145,167)(146,225)(147,224)(148,223)(149,222)(150,221)(151,226)(152,230)(153,229)(154,228)(155,227)(156,159)(157,158)(161,214)(162,213)(163,212)(164,211)(165,215)(171,200)(172,199)(173,198)(174,197)(175,196)(176,201)(177,205)(178,204)(179,203)(180,202)(186,189)(187,188)(191,241)(192,245)(193,244)(194,243)(195,242)(206,234)(207,233)(208,232)(209,231)(210,235)(217,220)(218,219)(246,250)(247,249);
s1 := Sym(250)!(  2,  5)(  3,  4)(  6, 41)(  7, 45)(  8, 44)(  9, 43)( 10, 42)( 11, 59)( 12, 58)( 13, 57)( 14, 56)( 15, 60)( 16,100)( 17, 99)( 18, 98)( 19, 97)( 20, 96)( 21,114)( 22,113)( 23,112)( 24,111)( 25,115)( 26, 61)( 27, 65)( 28, 64)( 29, 63)( 30, 62)( 31, 78)( 32, 77)( 33, 76)( 34, 80)( 35, 79)( 36,118)( 37,117)( 38,116)( 39,120)( 40,119)( 46, 47)( 48, 50)( 51,125)( 52,124)( 53,123)( 54,122)( 55,121)( 67, 70)( 68, 69)( 71, 84)( 72, 83)( 73, 82)( 74, 81)( 75, 85)( 86, 88)( 89, 90)( 91,105)( 92,104)( 93,103)( 94,102)( 95,101)(106,108)(109,110)(127,130)(128,129)(131,166)(132,170)(133,169)(134,168)(135,167)(136,184)(137,183)(138,182)(139,181)(140,185)(141,225)(142,224)(143,223)(144,222)(145,221)(146,239)(147,238)(148,237)(149,236)(150,240)(151,186)(152,190)(153,189)(154,188)(155,187)(156,203)(157,202)(158,201)(159,205)(160,204)(161,243)(162,242)(163,241)(164,245)(165,244)(171,172)(173,175)(176,250)(177,249)(178,248)(179,247)(180,246)(192,195)(193,194)(196,209)(197,208)(198,207)(199,206)(200,210)(211,213)(214,215)(216,230)(217,229)(218,228)(219,227)(220,226)(231,233)(234,235);
s2 := Sym(250)!(  1,216)(  2,220)(  3,219)(  4,218)(  5,217)(  6,162)(  7,161)(  8,165)(  9,164)( 10,163)( 11,235)( 12,234)( 13,233)( 14,232)( 15,231)( 16,180)( 17,179)( 18,178)( 19,177)( 20,176)( 21,147)( 22,146)( 23,150)( 24,149)( 25,148)( 26,243)( 27,242)( 28,241)( 29,245)( 30,244)( 31,189)( 32,188)( 33,187)( 34,186)( 35,190)( 36,132)( 37,131)( 38,135)( 39,134)( 40,133)( 41,202)( 42,201)( 43,205)( 44,204)( 45,203)( 46,174)( 47,173)( 48,172)( 49,171)( 50,175)( 51,145)( 52,144)( 53,143)( 54,142)( 55,141)( 56,211)( 57,215)( 58,214)( 59,213)( 60,212)( 61,159)( 62,158)( 63,157)( 64,156)( 65,160)( 66,229)( 67,228)( 68,227)( 69,226)( 70,230)( 71,196)( 72,200)( 73,199)( 74,198)( 75,197)( 76,167)( 77,166)( 78,170)( 79,169)( 80,168)( 81,238)( 82,237)( 83,236)( 84,240)( 85,239)( 86,181)( 87,185)( 88,184)( 89,183)( 90,182)( 91,126)( 92,130)( 93,129)( 94,128)( 95,127)( 96,223)( 97,222)( 98,221)( 99,225)(100,224)(101,194)(102,193)(103,192)(104,191)(105,195)(106,140)(107,139)(108,138)(109,137)(110,136)(111,208)(112,207)(113,206)(114,210)(115,209)(116,153)(117,152)(118,151)(119,155)(120,154)(121,250)(122,249)(123,248)(124,247)(125,246);
poly := sub<Sym(250)|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, 
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2*s1, 
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1 >;
References : None.
to this polytope

Twisty Puzzle