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Polytope of Type {12,78}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,78}*1872c
if this polytope has a name.
Group : SmallGroup(1872,922)
Rank : 3
Schlafli Type : {12,78}
Number of vertices, edges, etc : 12, 468, 78
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 : {6,78}*936c
   3-fold quotients : {4,78}*624a
   4-fold quotients : {6,39}*468
   6-fold quotients : {2,78}*312
   9-fold quotients : {4,26}*208
   12-fold quotients : {2,39}*156
   13-fold quotients : {12,6}*144c
   18-fold quotients : {2,26}*104
   26-fold quotients : {6,6}*72b
   36-fold quotients : {2,13}*52
   39-fold quotients : {4,6}*48a
   52-fold quotients : {6,3}*36
   78-fold quotients : {2,6}*24
   117-fold quotients : {4,2}*16
   156-fold quotients : {2,3}*12
   234-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 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);;
s1 := (  1,274)(  2,286)(  3,285)(  4,284)(  5,283)(  6,282)(  7,281)(  8,280)
(  9,279)( 10,278)( 11,277)( 12,276)( 13,275)( 14,300)( 15,312)( 16,311)
( 17,310)( 18,309)( 19,308)( 20,307)( 21,306)( 22,305)( 23,304)( 24,303)
( 25,302)( 26,301)( 27,287)( 28,299)( 29,298)( 30,297)( 31,296)( 32,295)
( 33,294)( 34,293)( 35,292)( 36,291)( 37,290)( 38,289)( 39,288)( 40,235)
( 41,247)( 42,246)( 43,245)( 44,244)( 45,243)( 46,242)( 47,241)( 48,240)
( 49,239)( 50,238)( 51,237)( 52,236)( 53,261)( 54,273)( 55,272)( 56,271)
( 57,270)( 58,269)( 59,268)( 60,267)( 61,266)( 62,265)( 63,264)( 64,263)
( 65,262)( 66,248)( 67,260)( 68,259)( 69,258)( 70,257)( 71,256)( 72,255)
( 73,254)( 74,253)( 75,252)( 76,251)( 77,250)( 78,249)( 79,313)( 80,325)
( 81,324)( 82,323)( 83,322)( 84,321)( 85,320)( 86,319)( 87,318)( 88,317)
( 89,316)( 90,315)( 91,314)( 92,339)( 93,351)( 94,350)( 95,349)( 96,348)
( 97,347)( 98,346)( 99,345)(100,344)(101,343)(102,342)(103,341)(104,340)
(105,326)(106,338)(107,337)(108,336)(109,335)(110,334)(111,333)(112,332)
(113,331)(114,330)(115,329)(116,328)(117,327)(118,391)(119,403)(120,402)
(121,401)(122,400)(123,399)(124,398)(125,397)(126,396)(127,395)(128,394)
(129,393)(130,392)(131,417)(132,429)(133,428)(134,427)(135,426)(136,425)
(137,424)(138,423)(139,422)(140,421)(141,420)(142,419)(143,418)(144,404)
(145,416)(146,415)(147,414)(148,413)(149,412)(150,411)(151,410)(152,409)
(153,408)(154,407)(155,406)(156,405)(157,352)(158,364)(159,363)(160,362)
(161,361)(162,360)(163,359)(164,358)(165,357)(166,356)(167,355)(168,354)
(169,353)(170,378)(171,390)(172,389)(173,388)(174,387)(175,386)(176,385)
(177,384)(178,383)(179,382)(180,381)(181,380)(182,379)(183,365)(184,377)
(185,376)(186,375)(187,374)(188,373)(189,372)(190,371)(191,370)(192,369)
(193,368)(194,367)(195,366)(196,430)(197,442)(198,441)(199,440)(200,439)
(201,438)(202,437)(203,436)(204,435)(205,434)(206,433)(207,432)(208,431)
(209,456)(210,468)(211,467)(212,466)(213,465)(214,464)(215,463)(216,462)
(217,461)(218,460)(219,459)(220,458)(221,457)(222,443)(223,455)(224,454)
(225,453)(226,452)(227,451)(228,450)(229,449)(230,448)(231,447)(232,446)
(233,445)(234,444);;
s2 := (  1, 15)(  2, 14)(  3, 26)(  4, 25)(  5, 24)(  6, 23)(  7, 22)(  8, 21)
(  9, 20)( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 27, 28)( 29, 39)( 30, 38)
( 31, 37)( 32, 36)( 33, 35)( 40, 93)( 41, 92)( 42,104)( 43,103)( 44,102)
( 45,101)( 46,100)( 47, 99)( 48, 98)( 49, 97)( 50, 96)( 51, 95)( 52, 94)
( 53, 80)( 54, 79)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 87)( 60, 86)
( 61, 85)( 62, 84)( 63, 83)( 64, 82)( 65, 81)( 66,106)( 67,105)( 68,117)
( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)( 76,109)
( 77,108)( 78,107)(118,132)(119,131)(120,143)(121,142)(122,141)(123,140)
(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)(144,145)
(146,156)(147,155)(148,154)(149,153)(150,152)(157,210)(158,209)(159,221)
(160,220)(161,219)(162,218)(163,217)(164,216)(165,215)(166,214)(167,213)
(168,212)(169,211)(170,197)(171,196)(172,208)(173,207)(174,206)(175,205)
(176,204)(177,203)(178,202)(179,201)(180,200)(181,199)(182,198)(183,223)
(184,222)(185,234)(186,233)(187,232)(188,231)(189,230)(190,229)(191,228)
(192,227)(193,226)(194,225)(195,224)(235,249)(236,248)(237,260)(238,259)
(239,258)(240,257)(241,256)(242,255)(243,254)(244,253)(245,252)(246,251)
(247,250)(261,262)(263,273)(264,272)(265,271)(266,270)(267,269)(274,327)
(275,326)(276,338)(277,337)(278,336)(279,335)(280,334)(281,333)(282,332)
(283,331)(284,330)(285,329)(286,328)(287,314)(288,313)(289,325)(290,324)
(291,323)(292,322)(293,321)(294,320)(295,319)(296,318)(297,317)(298,316)
(299,315)(300,340)(301,339)(302,351)(303,350)(304,349)(305,348)(306,347)
(307,346)(308,345)(309,344)(310,343)(311,342)(312,341)(352,366)(353,365)
(354,377)(355,376)(356,375)(357,374)(358,373)(359,372)(360,371)(361,370)
(362,369)(363,368)(364,367)(378,379)(380,390)(381,389)(382,388)(383,387)
(384,386)(391,444)(392,443)(393,455)(394,454)(395,453)(396,452)(397,451)
(398,450)(399,449)(400,448)(401,447)(402,446)(403,445)(404,431)(405,430)
(406,442)(407,441)(408,440)(409,439)(410,438)(411,437)(412,436)(413,435)
(414,434)(415,433)(416,432)(417,457)(418,456)(419,468)(420,467)(421,466)
(422,465)(423,464)(424,463)(425,462)(426,461)(427,460)(428,459)(429,458);;
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*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, 
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*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(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);
s1 := Sym(468)!(  1,274)(  2,286)(  3,285)(  4,284)(  5,283)(  6,282)(  7,281)
(  8,280)(  9,279)( 10,278)( 11,277)( 12,276)( 13,275)( 14,300)( 15,312)
( 16,311)( 17,310)( 18,309)( 19,308)( 20,307)( 21,306)( 22,305)( 23,304)
( 24,303)( 25,302)( 26,301)( 27,287)( 28,299)( 29,298)( 30,297)( 31,296)
( 32,295)( 33,294)( 34,293)( 35,292)( 36,291)( 37,290)( 38,289)( 39,288)
( 40,235)( 41,247)( 42,246)( 43,245)( 44,244)( 45,243)( 46,242)( 47,241)
( 48,240)( 49,239)( 50,238)( 51,237)( 52,236)( 53,261)( 54,273)( 55,272)
( 56,271)( 57,270)( 58,269)( 59,268)( 60,267)( 61,266)( 62,265)( 63,264)
( 64,263)( 65,262)( 66,248)( 67,260)( 68,259)( 69,258)( 70,257)( 71,256)
( 72,255)( 73,254)( 74,253)( 75,252)( 76,251)( 77,250)( 78,249)( 79,313)
( 80,325)( 81,324)( 82,323)( 83,322)( 84,321)( 85,320)( 86,319)( 87,318)
( 88,317)( 89,316)( 90,315)( 91,314)( 92,339)( 93,351)( 94,350)( 95,349)
( 96,348)( 97,347)( 98,346)( 99,345)(100,344)(101,343)(102,342)(103,341)
(104,340)(105,326)(106,338)(107,337)(108,336)(109,335)(110,334)(111,333)
(112,332)(113,331)(114,330)(115,329)(116,328)(117,327)(118,391)(119,403)
(120,402)(121,401)(122,400)(123,399)(124,398)(125,397)(126,396)(127,395)
(128,394)(129,393)(130,392)(131,417)(132,429)(133,428)(134,427)(135,426)
(136,425)(137,424)(138,423)(139,422)(140,421)(141,420)(142,419)(143,418)
(144,404)(145,416)(146,415)(147,414)(148,413)(149,412)(150,411)(151,410)
(152,409)(153,408)(154,407)(155,406)(156,405)(157,352)(158,364)(159,363)
(160,362)(161,361)(162,360)(163,359)(164,358)(165,357)(166,356)(167,355)
(168,354)(169,353)(170,378)(171,390)(172,389)(173,388)(174,387)(175,386)
(176,385)(177,384)(178,383)(179,382)(180,381)(181,380)(182,379)(183,365)
(184,377)(185,376)(186,375)(187,374)(188,373)(189,372)(190,371)(191,370)
(192,369)(193,368)(194,367)(195,366)(196,430)(197,442)(198,441)(199,440)
(200,439)(201,438)(202,437)(203,436)(204,435)(205,434)(206,433)(207,432)
(208,431)(209,456)(210,468)(211,467)(212,466)(213,465)(214,464)(215,463)
(216,462)(217,461)(218,460)(219,459)(220,458)(221,457)(222,443)(223,455)
(224,454)(225,453)(226,452)(227,451)(228,450)(229,449)(230,448)(231,447)
(232,446)(233,445)(234,444);
s2 := Sym(468)!(  1, 15)(  2, 14)(  3, 26)(  4, 25)(  5, 24)(  6, 23)(  7, 22)
(  8, 21)(  9, 20)( 10, 19)( 11, 18)( 12, 17)( 13, 16)( 27, 28)( 29, 39)
( 30, 38)( 31, 37)( 32, 36)( 33, 35)( 40, 93)( 41, 92)( 42,104)( 43,103)
( 44,102)( 45,101)( 46,100)( 47, 99)( 48, 98)( 49, 97)( 50, 96)( 51, 95)
( 52, 94)( 53, 80)( 54, 79)( 55, 91)( 56, 90)( 57, 89)( 58, 88)( 59, 87)
( 60, 86)( 61, 85)( 62, 84)( 63, 83)( 64, 82)( 65, 81)( 66,106)( 67,105)
( 68,117)( 69,116)( 70,115)( 71,114)( 72,113)( 73,112)( 74,111)( 75,110)
( 76,109)( 77,108)( 78,107)(118,132)(119,131)(120,143)(121,142)(122,141)
(123,140)(124,139)(125,138)(126,137)(127,136)(128,135)(129,134)(130,133)
(144,145)(146,156)(147,155)(148,154)(149,153)(150,152)(157,210)(158,209)
(159,221)(160,220)(161,219)(162,218)(163,217)(164,216)(165,215)(166,214)
(167,213)(168,212)(169,211)(170,197)(171,196)(172,208)(173,207)(174,206)
(175,205)(176,204)(177,203)(178,202)(179,201)(180,200)(181,199)(182,198)
(183,223)(184,222)(185,234)(186,233)(187,232)(188,231)(189,230)(190,229)
(191,228)(192,227)(193,226)(194,225)(195,224)(235,249)(236,248)(237,260)
(238,259)(239,258)(240,257)(241,256)(242,255)(243,254)(244,253)(245,252)
(246,251)(247,250)(261,262)(263,273)(264,272)(265,271)(266,270)(267,269)
(274,327)(275,326)(276,338)(277,337)(278,336)(279,335)(280,334)(281,333)
(282,332)(283,331)(284,330)(285,329)(286,328)(287,314)(288,313)(289,325)
(290,324)(291,323)(292,322)(293,321)(294,320)(295,319)(296,318)(297,317)
(298,316)(299,315)(300,340)(301,339)(302,351)(303,350)(304,349)(305,348)
(306,347)(307,346)(308,345)(309,344)(310,343)(311,342)(312,341)(352,366)
(353,365)(354,377)(355,376)(356,375)(357,374)(358,373)(359,372)(360,371)
(361,370)(362,369)(363,368)(364,367)(378,379)(380,390)(381,389)(382,388)
(383,387)(384,386)(391,444)(392,443)(393,455)(394,454)(395,453)(396,452)
(397,451)(398,450)(399,449)(400,448)(401,447)(402,446)(403,445)(404,431)
(405,430)(406,442)(407,441)(408,440)(409,439)(410,438)(411,437)(412,436)
(413,435)(414,434)(415,433)(416,432)(417,457)(418,456)(419,468)(420,467)
(421,466)(422,465)(423,464)(424,463)(425,462)(426,461)(427,460)(428,459)
(429,458);
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*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, 
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*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.
to this polytope