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}*1600
Also Known As : {4,20}4. if this polytope has another name.
Group : SmallGroup(1600,6624)
Rank : 3
Schlafli Type : {4,20}
Number of vertices, edges, etc : 40, 400, 200
Order of s0s1s2 : 4
Order of s0s1s2s1 : 20
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
   Halving Operation
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,20}*800
   4-fold quotients : {4,10}*400
   8-fold quotients : {4,10}*200
   25-fold quotients : {4,4}*64
   50-fold quotients : {4,4}*32
   100-fold quotients : {2,4}*16, {4,2}*16
   200-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*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      100 facets:
         100 of {4}*8
      30 vertex figures:
         20 of {10}*20
         10 of {20}*40
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1> of order 4.
      50 facets:
         50 of {4}*8
      16 vertex figures:
         9 of {10}*20
         5 of {20}*40
         2 of {5}*10
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1> of order 5.
      40 facets:
         40 of {4}*8
      8 vertex figures:
         8 of {20}*40
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2> of order 5.
      40 facets:
         40 of {4}*8
      24 vertex figures:
         20 of {4}*8
         4 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 5.
      40 facets:
         40 of {4}*8
      8 vertex figures:
         8 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 10.
      20 facets:
         20 of {4}*8
      6 vertex figures:
         4 of {10}*20
         2 of {20}*40
   P/N, where N=<s1*s2*s1*s2> of order 10.
      20 facets:
         20 of {4}*8
      22 vertex figures:
         20 of {2}*4
         2 of {20}*40
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1, s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2> of order 10.
      20 facets:
         20 of {4}*8
      6 vertex figures:
         4 of {10}*20
         2 of {20}*40
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2> of order 10.
      20 facets:
         20 of {4}*8
      14 vertex figures:
         10 of {4}*8
         4 of {10}*20
   P/N, where N=<s1*s2*s1*s2*s1*s2*s1*s2, s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1> of order 20.
      10 facets:
         10 of {4}*8
      8 vertex figures:
         5 of {4}*8
         2 of {5}*10
         1 of {10}*20

Permutation Representation (GAP) : 
s0 := (  2, 12)(  3, 23)(  4,  9)(  5, 20)(  6, 21)(  8, 18)( 10, 15)( 11, 16)( 14, 24)( 17, 22)( 27, 37)( 28, 48)( 29, 34)( 30, 45)( 31, 46)( 33, 43)( 35, 40)( 36, 41)( 39, 49)( 42, 47)( 51, 76)( 52, 87)( 53, 98)( 54, 84)( 55, 95)( 56, 96)( 57, 82)( 58, 93)( 59, 79)( 60, 90)( 61, 91)( 62, 77)( 63, 88)( 64, 99)( 65, 85)( 66, 86)( 67, 97)( 68, 83)( 69, 94)( 70, 80)( 71, 81)( 72, 92)( 73, 78)( 74, 89)( 75,100)(102,112)(103,123)(104,109)(105,120)(106,121)(108,118)(110,115)(111,116)(114,124)(117,122)(127,137)(128,148)(129,134)(130,145)(131,146)(133,143)(135,140)(136,141)(139,149)(142,147)(151,176)(152,187)(153,198)(154,184)(155,195)(156,196)(157,182)(158,193)(159,179)(160,190)(161,191)(162,177)(163,188)(164,199)(165,185)(166,186)(167,197)(168,183)(169,194)(170,180)(171,181)(172,192)(173,178)(174,189)(175,200);;
s1 := (  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)( 11, 25)( 15, 17)( 19, 21)( 27, 34)( 28, 37)( 29, 45)( 30, 48)( 31, 38)( 32, 41)( 33, 49)( 36, 50)( 40, 42)( 44, 46)( 52, 59)( 53, 62)( 54, 70)( 55, 73)( 56, 63)( 57, 66)( 58, 74)( 61, 75)( 65, 67)( 69, 71)( 77, 84)( 78, 87)( 79, 95)( 80, 98)( 81, 88)( 82, 91)( 83, 99)( 86,100)( 90, 92)( 94, 96)(101,151)(102,159)(103,162)(104,170)(105,173)(106,163)(107,166)(108,174)(109,152)(110,160)(111,175)(112,153)(113,156)(114,164)(115,167)(116,157)(117,165)(118,168)(119,171)(120,154)(121,169)(122,172)(123,155)(124,158)(125,161)(126,176)(127,184)(128,187)(129,195)(130,198)(131,188)(132,191)(133,199)(134,177)(135,185)(136,200)(137,178)(138,181)(139,189)(140,192)(141,182)(142,190)(143,193)(144,196)(145,179)(146,194)(147,197)(148,180)(149,183)(150,186);;
s2 := (  1,132)(  2,131)(  3,135)(  4,134)(  5,133)(  6,127)(  7,126)(  8,130)(  9,129)( 10,128)( 11,147)( 12,146)( 13,150)( 14,149)( 15,148)( 16,142)( 17,141)( 18,145)( 19,144)( 20,143)( 21,137)( 22,136)( 23,140)( 24,139)( 25,138)( 26,107)( 27,106)( 28,110)( 29,109)( 30,108)( 31,102)( 32,101)( 33,105)( 34,104)( 35,103)( 36,122)( 37,121)( 38,125)( 39,124)( 40,123)( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 46,112)( 47,111)( 48,115)( 49,114)( 50,113)( 51,182)( 52,181)( 53,185)( 54,184)( 55,183)( 56,177)( 57,176)( 58,180)( 59,179)( 60,178)( 61,197)( 62,196)( 63,200)( 64,199)( 65,198)( 66,192)( 67,191)( 68,195)( 69,194)( 70,193)( 71,187)( 72,186)( 73,190)( 74,189)( 75,188)( 76,157)( 77,156)( 78,160)( 79,159)( 80,158)( 81,152)( 82,151)( 83,155)( 84,154)( 85,153)( 86,172)( 87,171)( 88,175)( 89,174)( 90,173)( 91,167)( 92,166)( 93,170)( 94,169)( 95,168)( 96,162)( 97,161)( 98,165)( 99,164)(100,163);;
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*s0*s1*s2*s0*s1*s2*s0*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(200)!(  2, 12)(  3, 23)(  4,  9)(  5, 20)(  6, 21)(  8, 18)( 10, 15)( 11, 16)( 14, 24)( 17, 22)( 27, 37)( 28, 48)( 29, 34)( 30, 45)( 31, 46)( 33, 43)( 35, 40)( 36, 41)( 39, 49)( 42, 47)( 51, 76)( 52, 87)( 53, 98)( 54, 84)( 55, 95)( 56, 96)( 57, 82)( 58, 93)( 59, 79)( 60, 90)( 61, 91)( 62, 77)( 63, 88)( 64, 99)( 65, 85)( 66, 86)( 67, 97)( 68, 83)( 69, 94)( 70, 80)( 71, 81)( 72, 92)( 73, 78)( 74, 89)( 75,100)(102,112)(103,123)(104,109)(105,120)(106,121)(108,118)(110,115)(111,116)(114,124)(117,122)(127,137)(128,148)(129,134)(130,145)(131,146)(133,143)(135,140)(136,141)(139,149)(142,147)(151,176)(152,187)(153,198)(154,184)(155,195)(156,196)(157,182)(158,193)(159,179)(160,190)(161,191)(162,177)(163,188)(164,199)(165,185)(166,186)(167,197)(168,183)(169,194)(170,180)(171,181)(172,192)(173,178)(174,189)(175,200);
s1 := Sym(200)!(  2,  9)(  3, 12)(  4, 20)(  5, 23)(  6, 13)(  7, 16)(  8, 24)( 11, 25)( 15, 17)( 19, 21)( 27, 34)( 28, 37)( 29, 45)( 30, 48)( 31, 38)( 32, 41)( 33, 49)( 36, 50)( 40, 42)( 44, 46)( 52, 59)( 53, 62)( 54, 70)( 55, 73)( 56, 63)( 57, 66)( 58, 74)( 61, 75)( 65, 67)( 69, 71)( 77, 84)( 78, 87)( 79, 95)( 80, 98)( 81, 88)( 82, 91)( 83, 99)( 86,100)( 90, 92)( 94, 96)(101,151)(102,159)(103,162)(104,170)(105,173)(106,163)(107,166)(108,174)(109,152)(110,160)(111,175)(112,153)(113,156)(114,164)(115,167)(116,157)(117,165)(118,168)(119,171)(120,154)(121,169)(122,172)(123,155)(124,158)(125,161)(126,176)(127,184)(128,187)(129,195)(130,198)(131,188)(132,191)(133,199)(134,177)(135,185)(136,200)(137,178)(138,181)(139,189)(140,192)(141,182)(142,190)(143,193)(144,196)(145,179)(146,194)(147,197)(148,180)(149,183)(150,186);
s2 := Sym(200)!(  1,132)(  2,131)(  3,135)(  4,134)(  5,133)(  6,127)(  7,126)(  8,130)(  9,129)( 10,128)( 11,147)( 12,146)( 13,150)( 14,149)( 15,148)( 16,142)( 17,141)( 18,145)( 19,144)( 20,143)( 21,137)( 22,136)( 23,140)( 24,139)( 25,138)( 26,107)( 27,106)( 28,110)( 29,109)( 30,108)( 31,102)( 32,101)( 33,105)( 34,104)( 35,103)( 36,122)( 37,121)( 38,125)( 39,124)( 40,123)( 41,117)( 42,116)( 43,120)( 44,119)( 45,118)( 46,112)( 47,111)( 48,115)( 49,114)( 50,113)( 51,182)( 52,181)( 53,185)( 54,184)( 55,183)( 56,177)( 57,176)( 58,180)( 59,179)( 60,178)( 61,197)( 62,196)( 63,200)( 64,199)( 65,198)( 66,192)( 67,191)( 68,195)( 69,194)( 70,193)( 71,187)( 72,186)( 73,190)( 74,189)( 75,188)( 76,157)( 77,156)( 78,160)( 79,159)( 80,158)( 81,152)( 82,151)( 83,155)( 84,154)( 85,153)( 86,172)( 87,171)( 88,175)( 89,174)( 90,173)( 91,167)( 92,166)( 93,170)( 94,169)( 95,168)( 96,162)( 97,161)( 98,165)( 99,164)(100,163);
poly := sub<Sym(200)|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*s0*s1*s2*s0*s1*s2*s0*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