Questions?
See the FAQ
or other info.

Polytope of Type {76,6}

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