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Polytope of Type {6,114}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,114}*1824
if this polytope has a name.
Group : SmallGroup(1824,1245)
Rank : 3
Schlafli Type : {6,114}
Number of vertices, edges, etc : 8, 456, 152
Order of s0s1s2 : 76
Order of s0s1s2s1 : 6
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 : {6,57}*912
   12-fold quotients : {2,38}*152
   19-fold quotients : {6,6}*96
   24-fold quotients : {2,19}*76
   38-fold quotients : {3,6}*48, {6,3}*48
   76-fold quotients : {3,3}*24
   228-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
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)( 55, 56)( 59, 60)( 63, 64)
( 67, 68)( 71, 72)( 75, 76)( 77,153)( 78,154)( 79,156)( 80,155)( 81,157)
( 82,158)( 83,160)( 84,159)( 85,161)( 86,162)( 87,164)( 88,163)( 89,165)
( 90,166)( 91,168)( 92,167)( 93,169)( 94,170)( 95,172)( 96,171)( 97,173)
( 98,174)( 99,176)(100,175)(101,177)(102,178)(103,180)(104,179)(105,181)
(106,182)(107,184)(108,183)(109,185)(110,186)(111,188)(112,187)(113,189)
(114,190)(115,192)(116,191)(117,193)(118,194)(119,196)(120,195)(121,197)
(122,198)(123,200)(124,199)(125,201)(126,202)(127,204)(128,203)(129,205)
(130,206)(131,208)(132,207)(133,209)(134,210)(135,212)(136,211)(137,213)
(138,214)(139,216)(140,215)(141,217)(142,218)(143,220)(144,219)(145,221)
(146,222)(147,224)(148,223)(149,225)(150,226)(151,228)(152,227)(231,232)
(235,236)(239,240)(243,244)(247,248)(251,252)(255,256)(259,260)(263,264)
(267,268)(271,272)(275,276)(279,280)(283,284)(287,288)(291,292)(295,296)
(299,300)(303,304)(305,381)(306,382)(307,384)(308,383)(309,385)(310,386)
(311,388)(312,387)(313,389)(314,390)(315,392)(316,391)(317,393)(318,394)
(319,396)(320,395)(321,397)(322,398)(323,400)(324,399)(325,401)(326,402)
(327,404)(328,403)(329,405)(330,406)(331,408)(332,407)(333,409)(334,410)
(335,412)(336,411)(337,413)(338,414)(339,416)(340,415)(341,417)(342,418)
(343,420)(344,419)(345,421)(346,422)(347,424)(348,423)(349,425)(350,426)
(351,428)(352,427)(353,429)(354,430)(355,432)(356,431)(357,433)(358,434)
(359,436)(360,435)(361,437)(362,438)(363,440)(364,439)(365,441)(366,442)
(367,444)(368,443)(369,445)(370,446)(371,448)(372,447)(373,449)(374,450)
(375,452)(376,451)(377,453)(378,454)(379,456)(380,455);;
s1 := (  1, 77)(  2, 80)(  3, 79)(  4, 78)(  5,149)(  6,152)(  7,151)(  8,150)
(  9,145)( 10,148)( 11,147)( 12,146)( 13,141)( 14,144)( 15,143)( 16,142)
( 17,137)( 18,140)( 19,139)( 20,138)( 21,133)( 22,136)( 23,135)( 24,134)
( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)( 32,126)
( 33,121)( 34,124)( 35,123)( 36,122)( 37,117)( 38,120)( 39,119)( 40,118)
( 41,113)( 42,116)( 43,115)( 44,114)( 45,109)( 46,112)( 47,111)( 48,110)
( 49,105)( 50,108)( 51,107)( 52,106)( 53,101)( 54,104)( 55,103)( 56,102)
( 57, 97)( 58,100)( 59, 99)( 60, 98)( 61, 93)( 62, 96)( 63, 95)( 64, 94)
( 65, 89)( 66, 92)( 67, 91)( 68, 90)( 69, 85)( 70, 88)( 71, 87)( 72, 86)
( 73, 81)( 74, 84)( 75, 83)( 76, 82)(154,156)(157,225)(158,228)(159,227)
(160,226)(161,221)(162,224)(163,223)(164,222)(165,217)(166,220)(167,219)
(168,218)(169,213)(170,216)(171,215)(172,214)(173,209)(174,212)(175,211)
(176,210)(177,205)(178,208)(179,207)(180,206)(181,201)(182,204)(183,203)
(184,202)(185,197)(186,200)(187,199)(188,198)(189,193)(190,196)(191,195)
(192,194)(229,305)(230,308)(231,307)(232,306)(233,377)(234,380)(235,379)
(236,378)(237,373)(238,376)(239,375)(240,374)(241,369)(242,372)(243,371)
(244,370)(245,365)(246,368)(247,367)(248,366)(249,361)(250,364)(251,363)
(252,362)(253,357)(254,360)(255,359)(256,358)(257,353)(258,356)(259,355)
(260,354)(261,349)(262,352)(263,351)(264,350)(265,345)(266,348)(267,347)
(268,346)(269,341)(270,344)(271,343)(272,342)(273,337)(274,340)(275,339)
(276,338)(277,333)(278,336)(279,335)(280,334)(281,329)(282,332)(283,331)
(284,330)(285,325)(286,328)(287,327)(288,326)(289,321)(290,324)(291,323)
(292,322)(293,317)(294,320)(295,319)(296,318)(297,313)(298,316)(299,315)
(300,314)(301,309)(302,312)(303,311)(304,310)(382,384)(385,453)(386,456)
(387,455)(388,454)(389,449)(390,452)(391,451)(392,450)(393,445)(394,448)
(395,447)(396,446)(397,441)(398,444)(399,443)(400,442)(401,437)(402,440)
(403,439)(404,438)(405,433)(406,436)(407,435)(408,434)(409,429)(410,432)
(411,431)(412,430)(413,425)(414,428)(415,427)(416,426)(417,421)(418,424)
(419,423)(420,422);;
s2 := (  1,234)(  2,233)(  3,235)(  4,236)(  5,230)(  6,229)(  7,231)(  8,232)
(  9,302)( 10,301)( 11,303)( 12,304)( 13,298)( 14,297)( 15,299)( 16,300)
( 17,294)( 18,293)( 19,295)( 20,296)( 21,290)( 22,289)( 23,291)( 24,292)
( 25,286)( 26,285)( 27,287)( 28,288)( 29,282)( 30,281)( 31,283)( 32,284)
( 33,278)( 34,277)( 35,279)( 36,280)( 37,274)( 38,273)( 39,275)( 40,276)
( 41,270)( 42,269)( 43,271)( 44,272)( 45,266)( 46,265)( 47,267)( 48,268)
( 49,262)( 50,261)( 51,263)( 52,264)( 53,258)( 54,257)( 55,259)( 56,260)
( 57,254)( 58,253)( 59,255)( 60,256)( 61,250)( 62,249)( 63,251)( 64,252)
( 65,246)( 66,245)( 67,247)( 68,248)( 69,242)( 70,241)( 71,243)( 72,244)
( 73,238)( 74,237)( 75,239)( 76,240)( 77,386)( 78,385)( 79,387)( 80,388)
( 81,382)( 82,381)( 83,383)( 84,384)( 85,454)( 86,453)( 87,455)( 88,456)
( 89,450)( 90,449)( 91,451)( 92,452)( 93,446)( 94,445)( 95,447)( 96,448)
( 97,442)( 98,441)( 99,443)(100,444)(101,438)(102,437)(103,439)(104,440)
(105,434)(106,433)(107,435)(108,436)(109,430)(110,429)(111,431)(112,432)
(113,426)(114,425)(115,427)(116,428)(117,422)(118,421)(119,423)(120,424)
(121,418)(122,417)(123,419)(124,420)(125,414)(126,413)(127,415)(128,416)
(129,410)(130,409)(131,411)(132,412)(133,406)(134,405)(135,407)(136,408)
(137,402)(138,401)(139,403)(140,404)(141,398)(142,397)(143,399)(144,400)
(145,394)(146,393)(147,395)(148,396)(149,390)(150,389)(151,391)(152,392)
(153,310)(154,309)(155,311)(156,312)(157,306)(158,305)(159,307)(160,308)
(161,378)(162,377)(163,379)(164,380)(165,374)(166,373)(167,375)(168,376)
(169,370)(170,369)(171,371)(172,372)(173,366)(174,365)(175,367)(176,368)
(177,362)(178,361)(179,363)(180,364)(181,358)(182,357)(183,359)(184,360)
(185,354)(186,353)(187,355)(188,356)(189,350)(190,349)(191,351)(192,352)
(193,346)(194,345)(195,347)(196,348)(197,342)(198,341)(199,343)(200,344)
(201,338)(202,337)(203,339)(204,340)(205,334)(206,333)(207,335)(208,336)
(209,330)(210,329)(211,331)(212,332)(213,326)(214,325)(215,327)(216,328)
(217,322)(218,321)(219,323)(220,324)(221,318)(222,317)(223,319)(224,320)
(225,314)(226,313)(227,315)(228,316);;
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*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*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*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(456)!(  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)( 55, 56)( 59, 60)
( 63, 64)( 67, 68)( 71, 72)( 75, 76)( 77,153)( 78,154)( 79,156)( 80,155)
( 81,157)( 82,158)( 83,160)( 84,159)( 85,161)( 86,162)( 87,164)( 88,163)
( 89,165)( 90,166)( 91,168)( 92,167)( 93,169)( 94,170)( 95,172)( 96,171)
( 97,173)( 98,174)( 99,176)(100,175)(101,177)(102,178)(103,180)(104,179)
(105,181)(106,182)(107,184)(108,183)(109,185)(110,186)(111,188)(112,187)
(113,189)(114,190)(115,192)(116,191)(117,193)(118,194)(119,196)(120,195)
(121,197)(122,198)(123,200)(124,199)(125,201)(126,202)(127,204)(128,203)
(129,205)(130,206)(131,208)(132,207)(133,209)(134,210)(135,212)(136,211)
(137,213)(138,214)(139,216)(140,215)(141,217)(142,218)(143,220)(144,219)
(145,221)(146,222)(147,224)(148,223)(149,225)(150,226)(151,228)(152,227)
(231,232)(235,236)(239,240)(243,244)(247,248)(251,252)(255,256)(259,260)
(263,264)(267,268)(271,272)(275,276)(279,280)(283,284)(287,288)(291,292)
(295,296)(299,300)(303,304)(305,381)(306,382)(307,384)(308,383)(309,385)
(310,386)(311,388)(312,387)(313,389)(314,390)(315,392)(316,391)(317,393)
(318,394)(319,396)(320,395)(321,397)(322,398)(323,400)(324,399)(325,401)
(326,402)(327,404)(328,403)(329,405)(330,406)(331,408)(332,407)(333,409)
(334,410)(335,412)(336,411)(337,413)(338,414)(339,416)(340,415)(341,417)
(342,418)(343,420)(344,419)(345,421)(346,422)(347,424)(348,423)(349,425)
(350,426)(351,428)(352,427)(353,429)(354,430)(355,432)(356,431)(357,433)
(358,434)(359,436)(360,435)(361,437)(362,438)(363,440)(364,439)(365,441)
(366,442)(367,444)(368,443)(369,445)(370,446)(371,448)(372,447)(373,449)
(374,450)(375,452)(376,451)(377,453)(378,454)(379,456)(380,455);
s1 := Sym(456)!(  1, 77)(  2, 80)(  3, 79)(  4, 78)(  5,149)(  6,152)(  7,151)
(  8,150)(  9,145)( 10,148)( 11,147)( 12,146)( 13,141)( 14,144)( 15,143)
( 16,142)( 17,137)( 18,140)( 19,139)( 20,138)( 21,133)( 22,136)( 23,135)
( 24,134)( 25,129)( 26,132)( 27,131)( 28,130)( 29,125)( 30,128)( 31,127)
( 32,126)( 33,121)( 34,124)( 35,123)( 36,122)( 37,117)( 38,120)( 39,119)
( 40,118)( 41,113)( 42,116)( 43,115)( 44,114)( 45,109)( 46,112)( 47,111)
( 48,110)( 49,105)( 50,108)( 51,107)( 52,106)( 53,101)( 54,104)( 55,103)
( 56,102)( 57, 97)( 58,100)( 59, 99)( 60, 98)( 61, 93)( 62, 96)( 63, 95)
( 64, 94)( 65, 89)( 66, 92)( 67, 91)( 68, 90)( 69, 85)( 70, 88)( 71, 87)
( 72, 86)( 73, 81)( 74, 84)( 75, 83)( 76, 82)(154,156)(157,225)(158,228)
(159,227)(160,226)(161,221)(162,224)(163,223)(164,222)(165,217)(166,220)
(167,219)(168,218)(169,213)(170,216)(171,215)(172,214)(173,209)(174,212)
(175,211)(176,210)(177,205)(178,208)(179,207)(180,206)(181,201)(182,204)
(183,203)(184,202)(185,197)(186,200)(187,199)(188,198)(189,193)(190,196)
(191,195)(192,194)(229,305)(230,308)(231,307)(232,306)(233,377)(234,380)
(235,379)(236,378)(237,373)(238,376)(239,375)(240,374)(241,369)(242,372)
(243,371)(244,370)(245,365)(246,368)(247,367)(248,366)(249,361)(250,364)
(251,363)(252,362)(253,357)(254,360)(255,359)(256,358)(257,353)(258,356)
(259,355)(260,354)(261,349)(262,352)(263,351)(264,350)(265,345)(266,348)
(267,347)(268,346)(269,341)(270,344)(271,343)(272,342)(273,337)(274,340)
(275,339)(276,338)(277,333)(278,336)(279,335)(280,334)(281,329)(282,332)
(283,331)(284,330)(285,325)(286,328)(287,327)(288,326)(289,321)(290,324)
(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)(297,313)(298,316)
(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)(382,384)(385,453)
(386,456)(387,455)(388,454)(389,449)(390,452)(391,451)(392,450)(393,445)
(394,448)(395,447)(396,446)(397,441)(398,444)(399,443)(400,442)(401,437)
(402,440)(403,439)(404,438)(405,433)(406,436)(407,435)(408,434)(409,429)
(410,432)(411,431)(412,430)(413,425)(414,428)(415,427)(416,426)(417,421)
(418,424)(419,423)(420,422);
s2 := Sym(456)!(  1,234)(  2,233)(  3,235)(  4,236)(  5,230)(  6,229)(  7,231)
(  8,232)(  9,302)( 10,301)( 11,303)( 12,304)( 13,298)( 14,297)( 15,299)
( 16,300)( 17,294)( 18,293)( 19,295)( 20,296)( 21,290)( 22,289)( 23,291)
( 24,292)( 25,286)( 26,285)( 27,287)( 28,288)( 29,282)( 30,281)( 31,283)
( 32,284)( 33,278)( 34,277)( 35,279)( 36,280)( 37,274)( 38,273)( 39,275)
( 40,276)( 41,270)( 42,269)( 43,271)( 44,272)( 45,266)( 46,265)( 47,267)
( 48,268)( 49,262)( 50,261)( 51,263)( 52,264)( 53,258)( 54,257)( 55,259)
( 56,260)( 57,254)( 58,253)( 59,255)( 60,256)( 61,250)( 62,249)( 63,251)
( 64,252)( 65,246)( 66,245)( 67,247)( 68,248)( 69,242)( 70,241)( 71,243)
( 72,244)( 73,238)( 74,237)( 75,239)( 76,240)( 77,386)( 78,385)( 79,387)
( 80,388)( 81,382)( 82,381)( 83,383)( 84,384)( 85,454)( 86,453)( 87,455)
( 88,456)( 89,450)( 90,449)( 91,451)( 92,452)( 93,446)( 94,445)( 95,447)
( 96,448)( 97,442)( 98,441)( 99,443)(100,444)(101,438)(102,437)(103,439)
(104,440)(105,434)(106,433)(107,435)(108,436)(109,430)(110,429)(111,431)
(112,432)(113,426)(114,425)(115,427)(116,428)(117,422)(118,421)(119,423)
(120,424)(121,418)(122,417)(123,419)(124,420)(125,414)(126,413)(127,415)
(128,416)(129,410)(130,409)(131,411)(132,412)(133,406)(134,405)(135,407)
(136,408)(137,402)(138,401)(139,403)(140,404)(141,398)(142,397)(143,399)
(144,400)(145,394)(146,393)(147,395)(148,396)(149,390)(150,389)(151,391)
(152,392)(153,310)(154,309)(155,311)(156,312)(157,306)(158,305)(159,307)
(160,308)(161,378)(162,377)(163,379)(164,380)(165,374)(166,373)(167,375)
(168,376)(169,370)(170,369)(171,371)(172,372)(173,366)(174,365)(175,367)
(176,368)(177,362)(178,361)(179,363)(180,364)(181,358)(182,357)(183,359)
(184,360)(185,354)(186,353)(187,355)(188,356)(189,350)(190,349)(191,351)
(192,352)(193,346)(194,345)(195,347)(196,348)(197,342)(198,341)(199,343)
(200,344)(201,338)(202,337)(203,339)(204,340)(205,334)(206,333)(207,335)
(208,336)(209,330)(210,329)(211,331)(212,332)(213,326)(214,325)(215,327)
(216,328)(217,322)(218,321)(219,323)(220,324)(221,318)(222,317)(223,319)
(224,320)(225,314)(226,313)(227,315)(228,316);
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*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*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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*s1*s2*s0*s1*s2*s1*s2*s0*s1*s2*s1 >; 
 
References : None.
to this polytope