Polytope of Type {8,52}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,52}*1664b
if this polytope has a name.
Group : SmallGroup(1664,6496)
Rank : 3
Schlafli Type : {8,52}
Number of vertices, edges, etc : 16, 416, 104
Order of s0s1s2 : 52
Order of s0s1s2s1 : 4
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,52}*832
   4-fold quotients : {4,52}*416
   8-fold quotients : {2,52}*208, {4,26}*208
   13-fold quotients : {8,4}*128b
   16-fold quotients : {2,26}*104
   26-fold quotients : {4,4}*64
   32-fold quotients : {2,13}*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=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      52 facets:
         52 of {8}*16
      8 vertex figures:
         8 of {52}*104

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