Polytope of Type {6,52}

Play with this polytope as a twisty puzzle

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

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

Twisty Puzzle