Polytope of Type {78,12}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {78,12}*1872c
if this polytope has a name.
Group : SmallGroup(1872,922)
Rank : 3
Schlafli Type : {78,12}
Number of vertices, edges, etc : 78, 468, 12
Order of s0s1s2 : 156
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
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 : {78,6}*936c
   3-fold quotients : {78,4}*624a
   4-fold quotients : {39,6}*468
   6-fold quotients : {78,2}*312
   9-fold quotients : {26,4}*208
   12-fold quotients : {39,2}*156
   13-fold quotients : {6,12}*144c
   18-fold quotients : {26,2}*104
   26-fold quotients : {6,6}*72c
   36-fold quotients : {13,2}*52
   39-fold quotients : {6,4}*48a
   52-fold quotients : {3,6}*36
   78-fold quotients : {6,2}*24
   117-fold quotients : {2,4}*16
   156-fold quotients : {3,2}*12
   234-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Irregular Quotients (of which this is a minimal cover):
   None.

Permutation Representation (GAP) : 
s0 := (  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 27)( 15, 39)( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 40, 79)( 41, 91)( 42, 90)( 43, 89)( 44, 88)( 45, 87)( 46, 86)( 47, 85)( 48, 84)( 49, 83)( 50, 82)( 51, 81)( 52, 80)( 53,105)( 54,117)( 55,116)( 56,115)( 57,114)( 58,113)( 59,112)( 60,111)( 61,110)( 62,109)( 63,108)( 64,107)( 65,106)( 66, 92)( 67,104)( 68,103)( 69,102)( 70,101)( 71,100)( 72, 99)( 73, 98)( 74, 97)( 75, 96)( 76, 95)( 77, 94)( 78, 93)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(131,144)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)(157,196)(158,208)(159,207)(160,206)(161,205)(162,204)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198)(169,197)(170,222)(171,234)(172,233)(173,232)(174,231)(175,230)(176,229)(177,228)(178,227)(179,226)(180,225)(181,224)(182,223)(183,209)(184,221)(185,220)(186,219)(187,218)(188,217)(189,216)(190,215)(191,214)(192,213)(193,212)(194,211)(195,210)(236,247)(237,246)(238,245)(239,244)(240,243)(241,242)(248,261)(249,273)(250,272)(251,271)(252,270)(253,269)(254,268)(255,267)(256,266)(257,265)(258,264)(259,263)(260,262)(274,313)(275,325)(276,324)(277,323)(278,322)(279,321)(280,320)(281,319)(282,318)(283,317)(284,316)(285,315)(286,314)(287,339)(288,351)(289,350)(290,349)(291,348)(292,347)(293,346)(294,345)(295,344)(296,343)(297,342)(298,341)(299,340)(300,326)(301,338)(302,337)(303,336)(304,335)(305,334)(306,333)(307,332)(308,331)(309,330)(310,329)(311,328)(312,327)(353,364)(354,363)(355,362)(356,361)(357,360)(358,359)(365,378)(366,390)(367,389)(368,388)(369,387)(370,386)(371,385)(372,384)(373,383)(374,382)(375,381)(376,380)(377,379)(391,430)(392,442)(393,441)(394,440)(395,439)(396,438)(397,437)(398,436)(399,435)(400,434)(401,433)(402,432)(403,431)(404,456)(405,468)(406,467)(407,466)(408,465)(409,464)(410,463)(411,462)(412,461)(413,460)(414,459)(415,458)(416,457)(417,443)(418,455)(419,454)(420,453)(421,452)(422,451)(423,450)(424,449)(425,448)(426,447)(427,446)(428,445)(429,444);;
s1 := (  1,288)(  2,287)(  3,299)(  4,298)(  5,297)(  6,296)(  7,295)(  8,294)(  9,293)( 10,292)( 11,291)( 12,290)( 13,289)( 14,275)( 15,274)( 16,286)( 17,285)( 18,284)( 19,283)( 20,282)( 21,281)( 22,280)( 23,279)( 24,278)( 25,277)( 26,276)( 27,301)( 28,300)( 29,312)( 30,311)( 31,310)( 32,309)( 33,308)( 34,307)( 35,306)( 36,305)( 37,304)( 38,303)( 39,302)( 40,249)( 41,248)( 42,260)( 43,259)( 44,258)( 45,257)( 46,256)( 47,255)( 48,254)( 49,253)( 50,252)( 51,251)( 52,250)( 53,236)( 54,235)( 55,247)( 56,246)( 57,245)( 58,244)( 59,243)( 60,242)( 61,241)( 62,240)( 63,239)( 64,238)( 65,237)( 66,262)( 67,261)( 68,273)( 69,272)( 70,271)( 71,270)( 72,269)( 73,268)( 74,267)( 75,266)( 76,265)( 77,264)( 78,263)( 79,327)( 80,326)( 81,338)( 82,337)( 83,336)( 84,335)( 85,334)( 86,333)( 87,332)( 88,331)( 89,330)( 90,329)( 91,328)( 92,314)( 93,313)( 94,325)( 95,324)( 96,323)( 97,322)( 98,321)( 99,320)(100,319)(101,318)(102,317)(103,316)(104,315)(105,340)(106,339)(107,351)(108,350)(109,349)(110,348)(111,347)(112,346)(113,345)(114,344)(115,343)(116,342)(117,341)(118,405)(119,404)(120,416)(121,415)(122,414)(123,413)(124,412)(125,411)(126,410)(127,409)(128,408)(129,407)(130,406)(131,392)(132,391)(133,403)(134,402)(135,401)(136,400)(137,399)(138,398)(139,397)(140,396)(141,395)(142,394)(143,393)(144,418)(145,417)(146,429)(147,428)(148,427)(149,426)(150,425)(151,424)(152,423)(153,422)(154,421)(155,420)(156,419)(157,366)(158,365)(159,377)(160,376)(161,375)(162,374)(163,373)(164,372)(165,371)(166,370)(167,369)(168,368)(169,367)(170,353)(171,352)(172,364)(173,363)(174,362)(175,361)(176,360)(177,359)(178,358)(179,357)(180,356)(181,355)(182,354)(183,379)(184,378)(185,390)(186,389)(187,388)(188,387)(189,386)(190,385)(191,384)(192,383)(193,382)(194,381)(195,380)(196,444)(197,443)(198,455)(199,454)(200,453)(201,452)(202,451)(203,450)(204,449)(205,448)(206,447)(207,446)(208,445)(209,431)(210,430)(211,442)(212,441)(213,440)(214,439)(215,438)(216,437)(217,436)(218,435)(219,434)(220,433)(221,432)(222,457)(223,456)(224,468)(225,467)(226,466)(227,465)(228,464)(229,463)(230,462)(231,461)(232,460)(233,459)(234,458);;
s2 := ( 40, 79)( 41, 80)( 42, 81)( 43, 82)( 44, 83)( 45, 84)( 46, 85)( 47, 86)( 48, 87)( 49, 88)( 50, 89)( 51, 90)( 52, 91)( 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)( 66,105)( 67,106)( 68,107)( 69,108)( 70,109)( 71,110)( 72,111)( 73,112)( 74,113)( 75,114)( 76,115)( 77,116)( 78,117)(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)(170,209)(171,210)(172,211)(173,212)(174,213)(175,214)(176,215)(177,216)(178,217)(179,218)(180,219)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(235,352)(236,353)(237,354)(238,355)(239,356)(240,357)(241,358)(242,359)(243,360)(244,361)(245,362)(246,363)(247,364)(248,365)(249,366)(250,367)(251,368)(252,369)(253,370)(254,371)(255,372)(256,373)(257,374)(258,375)(259,376)(260,377)(261,378)(262,379)(263,380)(264,381)(265,382)(266,383)(267,384)(268,385)(269,386)(270,387)(271,388)(272,389)(273,390)(274,430)(275,431)(276,432)(277,433)(278,434)(279,435)(280,436)(281,437)(282,438)(283,439)(284,440)(285,441)(286,442)(287,443)(288,444)(289,445)(290,446)(291,447)(292,448)(293,449)(294,450)(295,451)(296,452)(297,453)(298,454)(299,455)(300,456)(301,457)(302,458)(303,459)(304,460)(305,461)(306,462)(307,463)(308,464)(309,465)(310,466)(311,467)(312,468)(313,391)(314,392)(315,393)(316,394)(317,395)(318,396)(319,397)(320,398)(321,399)(322,400)(323,401)(324,402)(325,403)(326,404)(327,405)(328,406)(329,407)(330,408)(331,409)(332,410)(333,411)(334,412)(335,413)(336,414)(337,415)(338,416)(339,417)(340,418)(341,419)(342,420)(343,421)(344,422)(345,423)(346,424)(347,425)(348,426)(349,427)(350,428)(351,429);;
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*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*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(468)!(  2, 13)(  3, 12)(  4, 11)(  5, 10)(  6,  9)(  7,  8)( 14, 27)( 15, 39)( 16, 38)( 17, 37)( 18, 36)( 19, 35)( 20, 34)( 21, 33)( 22, 32)( 23, 31)( 24, 30)( 25, 29)( 26, 28)( 40, 79)( 41, 91)( 42, 90)( 43, 89)( 44, 88)( 45, 87)( 46, 86)( 47, 85)( 48, 84)( 49, 83)( 50, 82)( 51, 81)( 52, 80)( 53,105)( 54,117)( 55,116)( 56,115)( 57,114)( 58,113)( 59,112)( 60,111)( 61,110)( 62,109)( 63,108)( 64,107)( 65,106)( 66, 92)( 67,104)( 68,103)( 69,102)( 70,101)( 71,100)( 72, 99)( 73, 98)( 74, 97)( 75, 96)( 76, 95)( 77, 94)( 78, 93)(119,130)(120,129)(121,128)(122,127)(123,126)(124,125)(131,144)(132,156)(133,155)(134,154)(135,153)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)(157,196)(158,208)(159,207)(160,206)(161,205)(162,204)(163,203)(164,202)(165,201)(166,200)(167,199)(168,198)(169,197)(170,222)(171,234)(172,233)(173,232)(174,231)(175,230)(176,229)(177,228)(178,227)(179,226)(180,225)(181,224)(182,223)(183,209)(184,221)(185,220)(186,219)(187,218)(188,217)(189,216)(190,215)(191,214)(192,213)(193,212)(194,211)(195,210)(236,247)(237,246)(238,245)(239,244)(240,243)(241,242)(248,261)(249,273)(250,272)(251,271)(252,270)(253,269)(254,268)(255,267)(256,266)(257,265)(258,264)(259,263)(260,262)(274,313)(275,325)(276,324)(277,323)(278,322)(279,321)(280,320)(281,319)(282,318)(283,317)(284,316)(285,315)(286,314)(287,339)(288,351)(289,350)(290,349)(291,348)(292,347)(293,346)(294,345)(295,344)(296,343)(297,342)(298,341)(299,340)(300,326)(301,338)(302,337)(303,336)(304,335)(305,334)(306,333)(307,332)(308,331)(309,330)(310,329)(311,328)(312,327)(353,364)(354,363)(355,362)(356,361)(357,360)(358,359)(365,378)(366,390)(367,389)(368,388)(369,387)(370,386)(371,385)(372,384)(373,383)(374,382)(375,381)(376,380)(377,379)(391,430)(392,442)(393,441)(394,440)(395,439)(396,438)(397,437)(398,436)(399,435)(400,434)(401,433)(402,432)(403,431)(404,456)(405,468)(406,467)(407,466)(408,465)(409,464)(410,463)(411,462)(412,461)(413,460)(414,459)(415,458)(416,457)(417,443)(418,455)(419,454)(420,453)(421,452)(422,451)(423,450)(424,449)(425,448)(426,447)(427,446)(428,445)(429,444);
s1 := Sym(468)!(  1,288)(  2,287)(  3,299)(  4,298)(  5,297)(  6,296)(  7,295)(  8,294)(  9,293)( 10,292)( 11,291)( 12,290)( 13,289)( 14,275)( 15,274)( 16,286)( 17,285)( 18,284)( 19,283)( 20,282)( 21,281)( 22,280)( 23,279)( 24,278)( 25,277)( 26,276)( 27,301)( 28,300)( 29,312)( 30,311)( 31,310)( 32,309)( 33,308)( 34,307)( 35,306)( 36,305)( 37,304)( 38,303)( 39,302)( 40,249)( 41,248)( 42,260)( 43,259)( 44,258)( 45,257)( 46,256)( 47,255)( 48,254)( 49,253)( 50,252)( 51,251)( 52,250)( 53,236)( 54,235)( 55,247)( 56,246)( 57,245)( 58,244)( 59,243)( 60,242)( 61,241)( 62,240)( 63,239)( 64,238)( 65,237)( 66,262)( 67,261)( 68,273)( 69,272)( 70,271)( 71,270)( 72,269)( 73,268)( 74,267)( 75,266)( 76,265)( 77,264)( 78,263)( 79,327)( 80,326)( 81,338)( 82,337)( 83,336)( 84,335)( 85,334)( 86,333)( 87,332)( 88,331)( 89,330)( 90,329)( 91,328)( 92,314)( 93,313)( 94,325)( 95,324)( 96,323)( 97,322)( 98,321)( 99,320)(100,319)(101,318)(102,317)(103,316)(104,315)(105,340)(106,339)(107,351)(108,350)(109,349)(110,348)(111,347)(112,346)(113,345)(114,344)(115,343)(116,342)(117,341)(118,405)(119,404)(120,416)(121,415)(122,414)(123,413)(124,412)(125,411)(126,410)(127,409)(128,408)(129,407)(130,406)(131,392)(132,391)(133,403)(134,402)(135,401)(136,400)(137,399)(138,398)(139,397)(140,396)(141,395)(142,394)(143,393)(144,418)(145,417)(146,429)(147,428)(148,427)(149,426)(150,425)(151,424)(152,423)(153,422)(154,421)(155,420)(156,419)(157,366)(158,365)(159,377)(160,376)(161,375)(162,374)(163,373)(164,372)(165,371)(166,370)(167,369)(168,368)(169,367)(170,353)(171,352)(172,364)(173,363)(174,362)(175,361)(176,360)(177,359)(178,358)(179,357)(180,356)(181,355)(182,354)(183,379)(184,378)(185,390)(186,389)(187,388)(188,387)(189,386)(190,385)(191,384)(192,383)(193,382)(194,381)(195,380)(196,444)(197,443)(198,455)(199,454)(200,453)(201,452)(202,451)(203,450)(204,449)(205,448)(206,447)(207,446)(208,445)(209,431)(210,430)(211,442)(212,441)(213,440)(214,439)(215,438)(216,437)(217,436)(218,435)(219,434)(220,433)(221,432)(222,457)(223,456)(224,468)(225,467)(226,466)(227,465)(228,464)(229,463)(230,462)(231,461)(232,460)(233,459)(234,458);
s2 := Sym(468)!( 40, 79)( 41, 80)( 42, 81)( 43, 82)( 44, 83)( 45, 84)( 46, 85)( 47, 86)( 48, 87)( 49, 88)( 50, 89)( 51, 90)( 52, 91)( 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)( 66,105)( 67,106)( 68,107)( 69,108)( 70,109)( 71,110)( 72,111)( 73,112)( 74,113)( 75,114)( 76,115)( 77,116)( 78,117)(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)(170,209)(171,210)(172,211)(173,212)(174,213)(175,214)(176,215)(177,216)(178,217)(179,218)(180,219)(181,220)(182,221)(183,222)(184,223)(185,224)(186,225)(187,226)(188,227)(189,228)(190,229)(191,230)(192,231)(193,232)(194,233)(195,234)(235,352)(236,353)(237,354)(238,355)(239,356)(240,357)(241,358)(242,359)(243,360)(244,361)(245,362)(246,363)(247,364)(248,365)(249,366)(250,367)(251,368)(252,369)(253,370)(254,371)(255,372)(256,373)(257,374)(258,375)(259,376)(260,377)(261,378)(262,379)(263,380)(264,381)(265,382)(266,383)(267,384)(268,385)(269,386)(270,387)(271,388)(272,389)(273,390)(274,430)(275,431)(276,432)(277,433)(278,434)(279,435)(280,436)(281,437)(282,438)(283,439)(284,440)(285,441)(286,442)(287,443)(288,444)(289,445)(290,446)(291,447)(292,448)(293,449)(294,450)(295,451)(296,452)(297,453)(298,454)(299,455)(300,456)(301,457)(302,458)(303,459)(304,460)(305,461)(306,462)(307,463)(308,464)(309,465)(310,466)(311,467)(312,468)(313,391)(314,392)(315,393)(316,394)(317,395)(318,396)(319,397)(320,398)(321,399)(322,400)(323,401)(324,402)(325,403)(326,404)(327,405)(328,406)(329,407)(330,408)(331,409)(332,410)(333,411)(334,412)(335,413)(336,414)(337,415)(338,416)(339,417)(340,418)(341,419)(342,420)(343,421)(344,422)(345,423)(346,424)(347,425)(348,426)(349,427)(350,428)(351,429);
poly := sub<Sym(468)|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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*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*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.
to this polytope

Twisty Puzzle