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