Polytope of Type {52,6}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {52,6}*1248
if this polytope has a name.
Group : SmallGroup(1248,1438)
Rank : 3
Schlafli Type : {52,6}
Number of vertices, edges, etc : 104, 312, 12
Order of s0s1s2 : 78
Order of s0s1s2s1 : 4
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 : {52,6}*624b
   4-fold quotients : {26,6}*312
   12-fold quotients : {26,2}*104
   13-fold quotients : {4,6}*96
   24-fold quotients : {13,2}*52
   26-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
   52-fold quotients : {4,3}*24, {2,6}*24
   104-fold quotients : {2,3}*12
   156-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*s2*s1*s2*s1*s0*s1*s2*s1*s2> of order 2.
      8 facets:
         4 of {26}*52
         4 of {52}*104
      52 vertex figures:
         52 of {6}*12
   P/N, where N=<s0*s1*s0*s1*s2*s1*s0*s1*s2> of order 2.
      6 facets:
         6 of {52}*104
      52 vertex figures:
         52 of {6}*12

Permutation Representation (GAP) : 
s0 := (  1,  3)(  2,  4)(  5, 51)(  6, 52)(  7, 49)(  8, 50)(  9, 47)( 10, 48)( 11, 45)( 12, 46)( 13, 43)( 14, 44)( 15, 41)( 16, 42)( 17, 39)( 18, 40)( 19, 37)( 20, 38)( 21, 35)( 22, 36)( 23, 33)( 24, 34)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 53, 55)( 54, 56)( 57,103)( 58,104)( 59,101)( 60,102)( 61, 99)( 62,100)( 63, 97)( 64, 98)( 65, 95)( 66, 96)( 67, 93)( 68, 94)( 69, 91)( 70, 92)( 71, 89)( 72, 90)( 73, 87)( 74, 88)( 75, 85)( 76, 86)( 77, 83)( 78, 84)( 79, 81)( 80, 82)(105,107)(106,108)(109,155)(110,156)(111,153)(112,154)(113,151)(114,152)(115,149)(116,150)(117,147)(118,148)(119,145)(120,146)(121,143)(122,144)(123,141)(124,142)(125,139)(126,140)(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)(157,159)(158,160)(161,207)(162,208)(163,205)(164,206)(165,203)(166,204)(167,201)(168,202)(169,199)(170,200)(171,197)(172,198)(173,195)(174,196)(175,193)(176,194)(177,191)(178,192)(179,189)(180,190)(181,187)(182,188)(183,185)(184,186)(209,211)(210,212)(213,259)(214,260)(215,257)(216,258)(217,255)(218,256)(219,253)(220,254)(221,251)(222,252)(223,249)(224,250)(225,247)(226,248)(227,245)(228,246)(229,243)(230,244)(231,241)(232,242)(233,239)(234,240)(235,237)(236,238)(261,263)(262,264)(265,311)(266,312)(267,309)(268,310)(269,307)(270,308)(271,305)(272,306)(273,303)(274,304)(275,301)(276,302)(277,299)(278,300)(279,297)(280,298)(281,295)(282,296)(283,293)(284,294)(285,291)(286,292)(287,289)(288,290);;
s1 := (  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 45)( 14, 46)( 15, 48)( 16, 47)( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 31, 32)( 53,109)( 54,110)( 55,112)( 56,111)( 57,105)( 58,106)( 59,108)( 60,107)( 61,153)( 62,154)( 63,156)( 64,155)( 65,149)( 66,150)( 67,152)( 68,151)( 69,145)( 70,146)( 71,148)( 72,147)( 73,141)( 74,142)( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,133)( 82,134)( 83,136)( 84,135)( 85,129)( 86,130)( 87,132)( 88,131)( 89,125)( 90,126)( 91,128)( 92,127)( 93,121)( 94,122)( 95,124)( 96,123)( 97,117)( 98,118)( 99,120)(100,119)(101,113)(102,114)(103,116)(104,115)(157,161)(158,162)(159,164)(160,163)(165,205)(166,206)(167,208)(168,207)(169,201)(170,202)(171,204)(172,203)(173,197)(174,198)(175,200)(176,199)(177,193)(178,194)(179,196)(180,195)(181,189)(182,190)(183,192)(184,191)(187,188)(209,265)(210,266)(211,268)(212,267)(213,261)(214,262)(215,264)(216,263)(217,309)(218,310)(219,312)(220,311)(221,305)(222,306)(223,308)(224,307)(225,301)(226,302)(227,304)(228,303)(229,297)(230,298)(231,300)(232,299)(233,293)(234,294)(235,296)(236,295)(237,289)(238,290)(239,292)(240,291)(241,285)(242,286)(243,288)(244,287)(245,281)(246,282)(247,284)(248,283)(249,277)(250,278)(251,280)(252,279)(253,273)(254,274)(255,276)(256,275)(257,269)(258,270)(259,272)(260,271);;
s2 := (  1,209)(  2,212)(  3,211)(  4,210)(  5,213)(  6,216)(  7,215)(  8,214)(  9,217)( 10,220)( 11,219)( 12,218)( 13,221)( 14,224)( 15,223)( 16,222)( 17,225)( 18,228)( 19,227)( 20,226)( 21,229)( 22,232)( 23,231)( 24,230)( 25,233)( 26,236)( 27,235)( 28,234)( 29,237)( 30,240)( 31,239)( 32,238)( 33,241)( 34,244)( 35,243)( 36,242)( 37,245)( 38,248)( 39,247)( 40,246)( 41,249)( 42,252)( 43,251)( 44,250)( 45,253)( 46,256)( 47,255)( 48,254)( 49,257)( 50,260)( 51,259)( 52,258)( 53,157)( 54,160)( 55,159)( 56,158)( 57,161)( 58,164)( 59,163)( 60,162)( 61,165)( 62,168)( 63,167)( 64,166)( 65,169)( 66,172)( 67,171)( 68,170)( 69,173)( 70,176)( 71,175)( 72,174)( 73,177)( 74,180)( 75,179)( 76,178)( 77,181)( 78,184)( 79,183)( 80,182)( 81,185)( 82,188)( 83,187)( 84,186)( 85,189)( 86,192)( 87,191)( 88,190)( 89,193)( 90,196)( 91,195)( 92,194)( 93,197)( 94,200)( 95,199)( 96,198)( 97,201)( 98,204)( 99,203)(100,202)(101,205)(102,208)(103,207)(104,206)(105,261)(106,264)(107,263)(108,262)(109,265)(110,268)(111,267)(112,266)(113,269)(114,272)(115,271)(116,270)(117,273)(118,276)(119,275)(120,274)(121,277)(122,280)(123,279)(124,278)(125,281)(126,284)(127,283)(128,282)(129,285)(130,288)(131,287)(132,286)(133,289)(134,292)(135,291)(136,290)(137,293)(138,296)(139,295)(140,294)(141,297)(142,300)(143,299)(144,298)(145,301)(146,304)(147,303)(148,302)(149,305)(150,308)(151,307)(152,306)(153,309)(154,312)(155,311)(156,310);;
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*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*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*s2*s1*s0*s1*s0 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(312)!(  1,  3)(  2,  4)(  5, 51)(  6, 52)(  7, 49)(  8, 50)(  9, 47)( 10, 48)( 11, 45)( 12, 46)( 13, 43)( 14, 44)( 15, 41)( 16, 42)( 17, 39)( 18, 40)( 19, 37)( 20, 38)( 21, 35)( 22, 36)( 23, 33)( 24, 34)( 25, 31)( 26, 32)( 27, 29)( 28, 30)( 53, 55)( 54, 56)( 57,103)( 58,104)( 59,101)( 60,102)( 61, 99)( 62,100)( 63, 97)( 64, 98)( 65, 95)( 66, 96)( 67, 93)( 68, 94)( 69, 91)( 70, 92)( 71, 89)( 72, 90)( 73, 87)( 74, 88)( 75, 85)( 76, 86)( 77, 83)( 78, 84)( 79, 81)( 80, 82)(105,107)(106,108)(109,155)(110,156)(111,153)(112,154)(113,151)(114,152)(115,149)(116,150)(117,147)(118,148)(119,145)(120,146)(121,143)(122,144)(123,141)(124,142)(125,139)(126,140)(127,137)(128,138)(129,135)(130,136)(131,133)(132,134)(157,159)(158,160)(161,207)(162,208)(163,205)(164,206)(165,203)(166,204)(167,201)(168,202)(169,199)(170,200)(171,197)(172,198)(173,195)(174,196)(175,193)(176,194)(177,191)(178,192)(179,189)(180,190)(181,187)(182,188)(183,185)(184,186)(209,211)(210,212)(213,259)(214,260)(215,257)(216,258)(217,255)(218,256)(219,253)(220,254)(221,251)(222,252)(223,249)(224,250)(225,247)(226,248)(227,245)(228,246)(229,243)(230,244)(231,241)(232,242)(233,239)(234,240)(235,237)(236,238)(261,263)(262,264)(265,311)(266,312)(267,309)(268,310)(269,307)(270,308)(271,305)(272,306)(273,303)(274,304)(275,301)(276,302)(277,299)(278,300)(279,297)(280,298)(281,295)(282,296)(283,293)(284,294)(285,291)(286,292)(287,289)(288,290);
s1 := Sym(312)!(  1,  5)(  2,  6)(  3,  8)(  4,  7)(  9, 49)( 10, 50)( 11, 52)( 12, 51)( 13, 45)( 14, 46)( 15, 48)( 16, 47)( 17, 41)( 18, 42)( 19, 44)( 20, 43)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 33)( 26, 34)( 27, 36)( 28, 35)( 31, 32)( 53,109)( 54,110)( 55,112)( 56,111)( 57,105)( 58,106)( 59,108)( 60,107)( 61,153)( 62,154)( 63,156)( 64,155)( 65,149)( 66,150)( 67,152)( 68,151)( 69,145)( 70,146)( 71,148)( 72,147)( 73,141)( 74,142)( 75,144)( 76,143)( 77,137)( 78,138)( 79,140)( 80,139)( 81,133)( 82,134)( 83,136)( 84,135)( 85,129)( 86,130)( 87,132)( 88,131)( 89,125)( 90,126)( 91,128)( 92,127)( 93,121)( 94,122)( 95,124)( 96,123)( 97,117)( 98,118)( 99,120)(100,119)(101,113)(102,114)(103,116)(104,115)(157,161)(158,162)(159,164)(160,163)(165,205)(166,206)(167,208)(168,207)(169,201)(170,202)(171,204)(172,203)(173,197)(174,198)(175,200)(176,199)(177,193)(178,194)(179,196)(180,195)(181,189)(182,190)(183,192)(184,191)(187,188)(209,265)(210,266)(211,268)(212,267)(213,261)(214,262)(215,264)(216,263)(217,309)(218,310)(219,312)(220,311)(221,305)(222,306)(223,308)(224,307)(225,301)(226,302)(227,304)(228,303)(229,297)(230,298)(231,300)(232,299)(233,293)(234,294)(235,296)(236,295)(237,289)(238,290)(239,292)(240,291)(241,285)(242,286)(243,288)(244,287)(245,281)(246,282)(247,284)(248,283)(249,277)(250,278)(251,280)(252,279)(253,273)(254,274)(255,276)(256,275)(257,269)(258,270)(259,272)(260,271);
s2 := Sym(312)!(  1,209)(  2,212)(  3,211)(  4,210)(  5,213)(  6,216)(  7,215)(  8,214)(  9,217)( 10,220)( 11,219)( 12,218)( 13,221)( 14,224)( 15,223)( 16,222)( 17,225)( 18,228)( 19,227)( 20,226)( 21,229)( 22,232)( 23,231)( 24,230)( 25,233)( 26,236)( 27,235)( 28,234)( 29,237)( 30,240)( 31,239)( 32,238)( 33,241)( 34,244)( 35,243)( 36,242)( 37,245)( 38,248)( 39,247)( 40,246)( 41,249)( 42,252)( 43,251)( 44,250)( 45,253)( 46,256)( 47,255)( 48,254)( 49,257)( 50,260)( 51,259)( 52,258)( 53,157)( 54,160)( 55,159)( 56,158)( 57,161)( 58,164)( 59,163)( 60,162)( 61,165)( 62,168)( 63,167)( 64,166)( 65,169)( 66,172)( 67,171)( 68,170)( 69,173)( 70,176)( 71,175)( 72,174)( 73,177)( 74,180)( 75,179)( 76,178)( 77,181)( 78,184)( 79,183)( 80,182)( 81,185)( 82,188)( 83,187)( 84,186)( 85,189)( 86,192)( 87,191)( 88,190)( 89,193)( 90,196)( 91,195)( 92,194)( 93,197)( 94,200)( 95,199)( 96,198)( 97,201)( 98,204)( 99,203)(100,202)(101,205)(102,208)(103,207)(104,206)(105,261)(106,264)(107,263)(108,262)(109,265)(110,268)(111,267)(112,266)(113,269)(114,272)(115,271)(116,270)(117,273)(118,276)(119,275)(120,274)(121,277)(122,280)(123,279)(124,278)(125,281)(126,284)(127,283)(128,282)(129,285)(130,288)(131,287)(132,286)(133,289)(134,292)(135,291)(136,290)(137,293)(138,296)(139,295)(140,294)(141,297)(142,300)(143,299)(144,298)(145,301)(146,304)(147,303)(148,302)(149,305)(150,308)(151,307)(152,306)(153,309)(154,312)(155,311)(156,310);
poly := sub<Sym(312)|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*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1, 
s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*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*s2*s1*s0*s1*s0 >;
References : None.
to this polytope

Twisty Puzzle