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Polytope of Type {12,76}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,76}*1824
Also Known As : {12,76|2}. if this polytope has another name.
Group : SmallGroup(1824,620)
Rank : 3
Schlafli Type : {12,76}
Number of vertices, edges, etc : 12, 456, 76
Order of s0s1s2 : 228
Order of s0s1s2s1 : 2
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 : {12,38}*912, {6,76}*912a
   3-fold quotients : {4,76}*608
   4-fold quotients : {6,38}*456
   6-fold quotients : {2,76}*304, {4,38}*304
   12-fold quotients : {2,38}*152
   19-fold quotients : {12,4}*96a
   24-fold quotients : {2,19}*76
   38-fold quotients : {12,2}*48, {6,4}*48a
   57-fold quotients : {4,4}*32
   76-fold quotients : {6,2}*24
   114-fold quotients : {2,4}*16, {4,2}*16
   152-fold quotients : {3,2}*12
   228-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := ( 20, 39)( 21, 40)( 22, 41)( 23, 42)( 24, 43)( 25, 44)( 26, 45)( 27, 46)
( 28, 47)( 29, 48)( 30, 49)( 31, 50)( 32, 51)( 33, 52)( 34, 53)( 35, 54)
( 36, 55)( 37, 56)( 38, 57)( 77, 96)( 78, 97)( 79, 98)( 80, 99)( 81,100)
( 82,101)( 83,102)( 84,103)( 85,104)( 86,105)( 87,106)( 88,107)( 89,108)
( 90,109)( 91,110)( 92,111)( 93,112)( 94,113)( 95,114)(134,153)(135,154)
(136,155)(137,156)(138,157)(139,158)(140,159)(141,160)(142,161)(143,162)
(144,163)(145,164)(146,165)(147,166)(148,167)(149,168)(150,169)(151,170)
(152,171)(191,210)(192,211)(193,212)(194,213)(195,214)(196,215)(197,216)
(198,217)(199,218)(200,219)(201,220)(202,221)(203,222)(204,223)(205,224)
(206,225)(207,226)(208,227)(209,228)(229,343)(230,344)(231,345)(232,346)
(233,347)(234,348)(235,349)(236,350)(237,351)(238,352)(239,353)(240,354)
(241,355)(242,356)(243,357)(244,358)(245,359)(246,360)(247,361)(248,381)
(249,382)(250,383)(251,384)(252,385)(253,386)(254,387)(255,388)(256,389)
(257,390)(258,391)(259,392)(260,393)(261,394)(262,395)(263,396)(264,397)
(265,398)(266,399)(267,362)(268,363)(269,364)(270,365)(271,366)(272,367)
(273,368)(274,369)(275,370)(276,371)(277,372)(278,373)(279,374)(280,375)
(281,376)(282,377)(283,378)(284,379)(285,380)(286,400)(287,401)(288,402)
(289,403)(290,404)(291,405)(292,406)(293,407)(294,408)(295,409)(296,410)
(297,411)(298,412)(299,413)(300,414)(301,415)(302,416)(303,417)(304,418)
(305,438)(306,439)(307,440)(308,441)(309,442)(310,443)(311,444)(312,445)
(313,446)(314,447)(315,448)(316,449)(317,450)(318,451)(319,452)(320,453)
(321,454)(322,455)(323,456)(324,419)(325,420)(326,421)(327,422)(328,423)
(329,424)(330,425)(331,426)(332,427)(333,428)(334,429)(335,430)(336,431)
(337,432)(338,433)(339,434)(340,435)(341,436)(342,437);;
s1 := (  1,248)(  2,266)(  3,265)(  4,264)(  5,263)(  6,262)(  7,261)(  8,260)
(  9,259)( 10,258)( 11,257)( 12,256)( 13,255)( 14,254)( 15,253)( 16,252)
( 17,251)( 18,250)( 19,249)( 20,229)( 21,247)( 22,246)( 23,245)( 24,244)
( 25,243)( 26,242)( 27,241)( 28,240)( 29,239)( 30,238)( 31,237)( 32,236)
( 33,235)( 34,234)( 35,233)( 36,232)( 37,231)( 38,230)( 39,267)( 40,285)
( 41,284)( 42,283)( 43,282)( 44,281)( 45,280)( 46,279)( 47,278)( 48,277)
( 49,276)( 50,275)( 51,274)( 52,273)( 53,272)( 54,271)( 55,270)( 56,269)
( 57,268)( 58,305)( 59,323)( 60,322)( 61,321)( 62,320)( 63,319)( 64,318)
( 65,317)( 66,316)( 67,315)( 68,314)( 69,313)( 70,312)( 71,311)( 72,310)
( 73,309)( 74,308)( 75,307)( 76,306)( 77,286)( 78,304)( 79,303)( 80,302)
( 81,301)( 82,300)( 83,299)( 84,298)( 85,297)( 86,296)( 87,295)( 88,294)
( 89,293)( 90,292)( 91,291)( 92,290)( 93,289)( 94,288)( 95,287)( 96,324)
( 97,342)( 98,341)( 99,340)(100,339)(101,338)(102,337)(103,336)(104,335)
(105,334)(106,333)(107,332)(108,331)(109,330)(110,329)(111,328)(112,327)
(113,326)(114,325)(115,362)(116,380)(117,379)(118,378)(119,377)(120,376)
(121,375)(122,374)(123,373)(124,372)(125,371)(126,370)(127,369)(128,368)
(129,367)(130,366)(131,365)(132,364)(133,363)(134,343)(135,361)(136,360)
(137,359)(138,358)(139,357)(140,356)(141,355)(142,354)(143,353)(144,352)
(145,351)(146,350)(147,349)(148,348)(149,347)(150,346)(151,345)(152,344)
(153,381)(154,399)(155,398)(156,397)(157,396)(158,395)(159,394)(160,393)
(161,392)(162,391)(163,390)(164,389)(165,388)(166,387)(167,386)(168,385)
(169,384)(170,383)(171,382)(172,419)(173,437)(174,436)(175,435)(176,434)
(177,433)(178,432)(179,431)(180,430)(181,429)(182,428)(183,427)(184,426)
(185,425)(186,424)(187,423)(188,422)(189,421)(190,420)(191,400)(192,418)
(193,417)(194,416)(195,415)(196,414)(197,413)(198,412)(199,411)(200,410)
(201,409)(202,408)(203,407)(204,406)(205,405)(206,404)(207,403)(208,402)
(209,401)(210,438)(211,456)(212,455)(213,454)(214,453)(215,452)(216,451)
(217,450)(218,449)(219,448)(220,447)(221,446)(222,445)(223,444)(224,443)
(225,442)(226,441)(227,440)(228,439);;
s2 := (  1,  2)(  3, 19)(  4, 18)(  5, 17)(  6, 16)(  7, 15)(  8, 14)(  9, 13)
( 10, 12)( 20, 21)( 22, 38)( 23, 37)( 24, 36)( 25, 35)( 26, 34)( 27, 33)
( 28, 32)( 29, 31)( 39, 40)( 41, 57)( 42, 56)( 43, 55)( 44, 54)( 45, 53)
( 46, 52)( 47, 51)( 48, 50)( 58, 59)( 60, 76)( 61, 75)( 62, 74)( 63, 73)
( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 77, 78)( 79, 95)( 80, 94)( 81, 93)
( 82, 92)( 83, 91)( 84, 90)( 85, 89)( 86, 88)( 96, 97)( 98,114)( 99,113)
(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(115,116)(117,133)
(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)(134,135)
(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)(143,145)
(153,154)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)(161,165)
(162,164)(172,173)(174,190)(175,189)(176,188)(177,187)(178,186)(179,185)
(180,184)(181,183)(191,192)(193,209)(194,208)(195,207)(196,206)(197,205)
(198,204)(199,203)(200,202)(210,211)(212,228)(213,227)(214,226)(215,225)
(216,224)(217,223)(218,222)(219,221)(229,287)(230,286)(231,304)(232,303)
(233,302)(234,301)(235,300)(236,299)(237,298)(238,297)(239,296)(240,295)
(241,294)(242,293)(243,292)(244,291)(245,290)(246,289)(247,288)(248,306)
(249,305)(250,323)(251,322)(252,321)(253,320)(254,319)(255,318)(256,317)
(257,316)(258,315)(259,314)(260,313)(261,312)(262,311)(263,310)(264,309)
(265,308)(266,307)(267,325)(268,324)(269,342)(270,341)(271,340)(272,339)
(273,338)(274,337)(275,336)(276,335)(277,334)(278,333)(279,332)(280,331)
(281,330)(282,329)(283,328)(284,327)(285,326)(343,401)(344,400)(345,418)
(346,417)(347,416)(348,415)(349,414)(350,413)(351,412)(352,411)(353,410)
(354,409)(355,408)(356,407)(357,406)(358,405)(359,404)(360,403)(361,402)
(362,420)(363,419)(364,437)(365,436)(366,435)(367,434)(368,433)(369,432)
(370,431)(371,430)(372,429)(373,428)(374,427)(375,426)(376,425)(377,424)
(378,423)(379,422)(380,421)(381,439)(382,438)(383,456)(384,455)(385,454)
(386,453)(387,452)(388,451)(389,450)(390,449)(391,448)(392,447)(393,446)
(394,445)(395,444)(396,443)(397,442)(398,441)(399,440);;
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*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, 
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(456)!( 20, 39)( 21, 40)( 22, 41)( 23, 42)( 24, 43)( 25, 44)( 26, 45)
( 27, 46)( 28, 47)( 29, 48)( 30, 49)( 31, 50)( 32, 51)( 33, 52)( 34, 53)
( 35, 54)( 36, 55)( 37, 56)( 38, 57)( 77, 96)( 78, 97)( 79, 98)( 80, 99)
( 81,100)( 82,101)( 83,102)( 84,103)( 85,104)( 86,105)( 87,106)( 88,107)
( 89,108)( 90,109)( 91,110)( 92,111)( 93,112)( 94,113)( 95,114)(134,153)
(135,154)(136,155)(137,156)(138,157)(139,158)(140,159)(141,160)(142,161)
(143,162)(144,163)(145,164)(146,165)(147,166)(148,167)(149,168)(150,169)
(151,170)(152,171)(191,210)(192,211)(193,212)(194,213)(195,214)(196,215)
(197,216)(198,217)(199,218)(200,219)(201,220)(202,221)(203,222)(204,223)
(205,224)(206,225)(207,226)(208,227)(209,228)(229,343)(230,344)(231,345)
(232,346)(233,347)(234,348)(235,349)(236,350)(237,351)(238,352)(239,353)
(240,354)(241,355)(242,356)(243,357)(244,358)(245,359)(246,360)(247,361)
(248,381)(249,382)(250,383)(251,384)(252,385)(253,386)(254,387)(255,388)
(256,389)(257,390)(258,391)(259,392)(260,393)(261,394)(262,395)(263,396)
(264,397)(265,398)(266,399)(267,362)(268,363)(269,364)(270,365)(271,366)
(272,367)(273,368)(274,369)(275,370)(276,371)(277,372)(278,373)(279,374)
(280,375)(281,376)(282,377)(283,378)(284,379)(285,380)(286,400)(287,401)
(288,402)(289,403)(290,404)(291,405)(292,406)(293,407)(294,408)(295,409)
(296,410)(297,411)(298,412)(299,413)(300,414)(301,415)(302,416)(303,417)
(304,418)(305,438)(306,439)(307,440)(308,441)(309,442)(310,443)(311,444)
(312,445)(313,446)(314,447)(315,448)(316,449)(317,450)(318,451)(319,452)
(320,453)(321,454)(322,455)(323,456)(324,419)(325,420)(326,421)(327,422)
(328,423)(329,424)(330,425)(331,426)(332,427)(333,428)(334,429)(335,430)
(336,431)(337,432)(338,433)(339,434)(340,435)(341,436)(342,437);
s1 := Sym(456)!(  1,248)(  2,266)(  3,265)(  4,264)(  5,263)(  6,262)(  7,261)
(  8,260)(  9,259)( 10,258)( 11,257)( 12,256)( 13,255)( 14,254)( 15,253)
( 16,252)( 17,251)( 18,250)( 19,249)( 20,229)( 21,247)( 22,246)( 23,245)
( 24,244)( 25,243)( 26,242)( 27,241)( 28,240)( 29,239)( 30,238)( 31,237)
( 32,236)( 33,235)( 34,234)( 35,233)( 36,232)( 37,231)( 38,230)( 39,267)
( 40,285)( 41,284)( 42,283)( 43,282)( 44,281)( 45,280)( 46,279)( 47,278)
( 48,277)( 49,276)( 50,275)( 51,274)( 52,273)( 53,272)( 54,271)( 55,270)
( 56,269)( 57,268)( 58,305)( 59,323)( 60,322)( 61,321)( 62,320)( 63,319)
( 64,318)( 65,317)( 66,316)( 67,315)( 68,314)( 69,313)( 70,312)( 71,311)
( 72,310)( 73,309)( 74,308)( 75,307)( 76,306)( 77,286)( 78,304)( 79,303)
( 80,302)( 81,301)( 82,300)( 83,299)( 84,298)( 85,297)( 86,296)( 87,295)
( 88,294)( 89,293)( 90,292)( 91,291)( 92,290)( 93,289)( 94,288)( 95,287)
( 96,324)( 97,342)( 98,341)( 99,340)(100,339)(101,338)(102,337)(103,336)
(104,335)(105,334)(106,333)(107,332)(108,331)(109,330)(110,329)(111,328)
(112,327)(113,326)(114,325)(115,362)(116,380)(117,379)(118,378)(119,377)
(120,376)(121,375)(122,374)(123,373)(124,372)(125,371)(126,370)(127,369)
(128,368)(129,367)(130,366)(131,365)(132,364)(133,363)(134,343)(135,361)
(136,360)(137,359)(138,358)(139,357)(140,356)(141,355)(142,354)(143,353)
(144,352)(145,351)(146,350)(147,349)(148,348)(149,347)(150,346)(151,345)
(152,344)(153,381)(154,399)(155,398)(156,397)(157,396)(158,395)(159,394)
(160,393)(161,392)(162,391)(163,390)(164,389)(165,388)(166,387)(167,386)
(168,385)(169,384)(170,383)(171,382)(172,419)(173,437)(174,436)(175,435)
(176,434)(177,433)(178,432)(179,431)(180,430)(181,429)(182,428)(183,427)
(184,426)(185,425)(186,424)(187,423)(188,422)(189,421)(190,420)(191,400)
(192,418)(193,417)(194,416)(195,415)(196,414)(197,413)(198,412)(199,411)
(200,410)(201,409)(202,408)(203,407)(204,406)(205,405)(206,404)(207,403)
(208,402)(209,401)(210,438)(211,456)(212,455)(213,454)(214,453)(215,452)
(216,451)(217,450)(218,449)(219,448)(220,447)(221,446)(222,445)(223,444)
(224,443)(225,442)(226,441)(227,440)(228,439);
s2 := Sym(456)!(  1,  2)(  3, 19)(  4, 18)(  5, 17)(  6, 16)(  7, 15)(  8, 14)
(  9, 13)( 10, 12)( 20, 21)( 22, 38)( 23, 37)( 24, 36)( 25, 35)( 26, 34)
( 27, 33)( 28, 32)( 29, 31)( 39, 40)( 41, 57)( 42, 56)( 43, 55)( 44, 54)
( 45, 53)( 46, 52)( 47, 51)( 48, 50)( 58, 59)( 60, 76)( 61, 75)( 62, 74)
( 63, 73)( 64, 72)( 65, 71)( 66, 70)( 67, 69)( 77, 78)( 79, 95)( 80, 94)
( 81, 93)( 82, 92)( 83, 91)( 84, 90)( 85, 89)( 86, 88)( 96, 97)( 98,114)
( 99,113)(100,112)(101,111)(102,110)(103,109)(104,108)(105,107)(115,116)
(117,133)(118,132)(119,131)(120,130)(121,129)(122,128)(123,127)(124,126)
(134,135)(136,152)(137,151)(138,150)(139,149)(140,148)(141,147)(142,146)
(143,145)(153,154)(155,171)(156,170)(157,169)(158,168)(159,167)(160,166)
(161,165)(162,164)(172,173)(174,190)(175,189)(176,188)(177,187)(178,186)
(179,185)(180,184)(181,183)(191,192)(193,209)(194,208)(195,207)(196,206)
(197,205)(198,204)(199,203)(200,202)(210,211)(212,228)(213,227)(214,226)
(215,225)(216,224)(217,223)(218,222)(219,221)(229,287)(230,286)(231,304)
(232,303)(233,302)(234,301)(235,300)(236,299)(237,298)(238,297)(239,296)
(240,295)(241,294)(242,293)(243,292)(244,291)(245,290)(246,289)(247,288)
(248,306)(249,305)(250,323)(251,322)(252,321)(253,320)(254,319)(255,318)
(256,317)(257,316)(258,315)(259,314)(260,313)(261,312)(262,311)(263,310)
(264,309)(265,308)(266,307)(267,325)(268,324)(269,342)(270,341)(271,340)
(272,339)(273,338)(274,337)(275,336)(276,335)(277,334)(278,333)(279,332)
(280,331)(281,330)(282,329)(283,328)(284,327)(285,326)(343,401)(344,400)
(345,418)(346,417)(347,416)(348,415)(349,414)(350,413)(351,412)(352,411)
(353,410)(354,409)(355,408)(356,407)(357,406)(358,405)(359,404)(360,403)
(361,402)(362,420)(363,419)(364,437)(365,436)(366,435)(367,434)(368,433)
(369,432)(370,431)(371,430)(372,429)(373,428)(374,427)(375,426)(376,425)
(377,424)(378,423)(379,422)(380,421)(381,439)(382,438)(383,456)(384,455)
(385,454)(386,453)(387,452)(388,451)(389,450)(390,449)(391,448)(392,447)
(393,446)(394,445)(395,444)(396,443)(397,442)(398,441)(399,440);
poly := sub<Sym(456)|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*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, 
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