Polytope of Type {52,4}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {52,4}*1664
if this polytope has a name.
Group : SmallGroup(1664,6495)
Rank : 3
Schlafli Type : {52,4}
Number of vertices, edges, etc : 208, 416, 16
Order of s0s1s2 : 104
Order of s0s1s2s1 : 4
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 : {52,4}*832
   4-fold quotients : {52,4}*416
   8-fold quotients : {52,2}*208, {26,4}*208
   13-fold quotients : {4,4}*128
   16-fold quotients : {26,2}*104
   26-fold quotients : {4,4}*64
   32-fold quotients : {13,2}*52
   52-fold quotients : {4,4}*32
   104-fold quotients : {2,4}*16, {4,2}*16
   208-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> of order 2.
      8 facets:
         8 of {52}*104
      130 vertex figures:
         52 of {2}*4
         78 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      8 facets:
         8 of {52}*104
      104 vertex figures:
         104 of {4}*8
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1> of order 2.
      8 facets:
         8 of {52}*104
      104 vertex figures:
         104 of {4}*8
   P/N, where N=<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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1> of order 2.
      10 facets:
         4 of {26}*52
         6 of {52}*104
      104 vertex figures:
         104 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s2*s1*s0*s1*s2*s1*s2> of order 2.
      8 facets:
         8 of {52}*104
      104 vertex figures:
         104 of {4}*8
   P/N, where N=<s1*s0*s1*s2*s1*s0*s1*s2, s0*s1*s2*s1*s0*s1*s2*s1*s2> of order 4.
      4 facets:
         4 of {52}*104
      52 vertex figures:
         52 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s1*s0*s1*s2*s1*s0*s2*s1*s0*s2> of order 4.
      4 facets:
         4 of {52}*104
      65 vertex figures:
         26 of {2}*4
         39 of {4}*8
   P/N, where N=<s1*s2*s1*s2, s0*s1*s2*s1*s0*s2> of order 4.
      4 facets:
         4 of {52}*104
      78 vertex figures:
         52 of {2}*4
         26 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*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> of order 4.
      6 facets:
         4 of {26}*52
         2 of {52}*104
      52 vertex figures:
         52 of {4}*8
   P/N, where N=<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*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> of order 4.
      5 facets:
         2 of {26}*52
         3 of {52}*104
      52 vertex figures:
         52 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 4.
      4 facets:
         4 of {52}*104
      52 vertex figures:
         52 of {4}*8
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*s2*s1*s0*s2*s1*s2> of order 4.
      4 facets:
         4 of {52}*104
      52 vertex figures:
         52 of {4}*8

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