Polytope of Type {74,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {74,12}*1776
Also Known As : {74,12|2}. if this polytope has another name.
Group : SmallGroup(1776,178)
Rank : 3
Schlafli Type : {74,12}
Number of vertices, edges, etc : 74, 444, 12
Order of s0s1s2 : 444
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 : {74,6}*888
3-fold quotients : {74,4}*592
6-fold quotients : {74,2}*296
12-fold quotients : {37,2}*148
37-fold quotients : {2,12}*48
74-fold quotients : {2,6}*24
111-fold quotients : {2,4}*16
148-fold quotients : {2,3}*12
222-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 37)( 3, 36)( 4, 35)( 5, 34)( 6, 33)( 7, 32)( 8, 31)( 9, 30)
( 10, 29)( 11, 28)( 12, 27)( 13, 26)( 14, 25)( 15, 24)( 16, 23)( 17, 22)
( 18, 21)( 19, 20)( 39, 74)( 40, 73)( 41, 72)( 42, 71)( 43, 70)( 44, 69)
( 45, 68)( 46, 67)( 47, 66)( 48, 65)( 49, 64)( 50, 63)( 51, 62)( 52, 61)
( 53, 60)( 54, 59)( 55, 58)( 56, 57)( 76,111)( 77,110)( 78,109)( 79,108)
( 80,107)( 81,106)( 82,105)( 83,104)( 84,103)( 85,102)( 86,101)( 87,100)
( 88, 99)( 89, 98)( 90, 97)( 91, 96)( 92, 95)( 93, 94)(113,148)(114,147)
(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)(122,139)
(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)(130,131)
(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)(157,178)
(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)(165,170)
(166,169)(167,168)(187,222)(188,221)(189,220)(190,219)(191,218)(192,217)
(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(199,210)(200,209)
(201,208)(202,207)(203,206)(204,205)(224,259)(225,258)(226,257)(227,256)
(228,255)(229,254)(230,253)(231,252)(232,251)(233,250)(234,249)(235,248)
(236,247)(237,246)(238,245)(239,244)(240,243)(241,242)(261,296)(262,295)
(263,294)(264,293)(265,292)(266,291)(267,290)(268,289)(269,288)(270,287)
(271,286)(272,285)(273,284)(274,283)(275,282)(276,281)(277,280)(278,279)
(298,333)(299,332)(300,331)(301,330)(302,329)(303,328)(304,327)(305,326)
(306,325)(307,324)(308,323)(309,322)(310,321)(311,320)(312,319)(313,318)
(314,317)(315,316)(335,370)(336,369)(337,368)(338,367)(339,366)(340,365)
(341,364)(342,363)(343,362)(344,361)(345,360)(346,359)(347,358)(348,357)
(349,356)(350,355)(351,354)(352,353)(372,407)(373,406)(374,405)(375,404)
(376,403)(377,402)(378,401)(379,400)(380,399)(381,398)(382,397)(383,396)
(384,395)(385,394)(386,393)(387,392)(388,391)(389,390)(409,444)(410,443)
(411,442)(412,441)(413,440)(414,439)(415,438)(416,437)(417,436)(418,435)
(419,434)(420,433)(421,432)(422,431)(423,430)(424,429)(425,428)(426,427);;
s1 := ( 1, 2)( 3, 37)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)( 9, 31)
( 10, 30)( 11, 29)( 12, 28)( 13, 27)( 14, 26)( 15, 25)( 16, 24)( 17, 23)
( 18, 22)( 19, 21)( 38, 76)( 39, 75)( 40,111)( 41,110)( 42,109)( 43,108)
( 44,107)( 45,106)( 46,105)( 47,104)( 48,103)( 49,102)( 50,101)( 51,100)
( 52, 99)( 53, 98)( 54, 97)( 55, 96)( 56, 95)( 57, 94)( 58, 93)( 59, 92)
( 60, 91)( 61, 90)( 62, 89)( 63, 88)( 64, 87)( 65, 86)( 66, 85)( 67, 84)
( 68, 83)( 69, 82)( 70, 81)( 71, 80)( 72, 79)( 73, 78)( 74, 77)(112,113)
(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)(121,141)
(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)(129,133)
(130,132)(149,187)(150,186)(151,222)(152,221)(153,220)(154,219)(155,218)
(156,217)(157,216)(158,215)(159,214)(160,213)(161,212)(162,211)(163,210)
(164,209)(165,208)(166,207)(167,206)(168,205)(169,204)(170,203)(171,202)
(172,201)(173,200)(174,199)(175,198)(176,197)(177,196)(178,195)(179,194)
(180,193)(181,192)(182,191)(183,190)(184,189)(185,188)(223,335)(224,334)
(225,370)(226,369)(227,368)(228,367)(229,366)(230,365)(231,364)(232,363)
(233,362)(234,361)(235,360)(236,359)(237,358)(238,357)(239,356)(240,355)
(241,354)(242,353)(243,352)(244,351)(245,350)(246,349)(247,348)(248,347)
(249,346)(250,345)(251,344)(252,343)(253,342)(254,341)(255,340)(256,339)
(257,338)(258,337)(259,336)(260,409)(261,408)(262,444)(263,443)(264,442)
(265,441)(266,440)(267,439)(268,438)(269,437)(270,436)(271,435)(272,434)
(273,433)(274,432)(275,431)(276,430)(277,429)(278,428)(279,427)(280,426)
(281,425)(282,424)(283,423)(284,422)(285,421)(286,420)(287,419)(288,418)
(289,417)(290,416)(291,415)(292,414)(293,413)(294,412)(295,411)(296,410)
(297,372)(298,371)(299,407)(300,406)(301,405)(302,404)(303,403)(304,402)
(305,401)(306,400)(307,399)(308,398)(309,397)(310,396)(311,395)(312,394)
(313,393)(314,392)(315,391)(316,390)(317,389)(318,388)(319,387)(320,386)
(321,385)(322,384)(323,383)(324,382)(325,381)(326,380)(327,379)(328,378)
(329,377)(330,376)(331,375)(332,374)(333,373);;
s2 := ( 1,260)( 2,261)( 3,262)( 4,263)( 5,264)( 6,265)( 7,266)( 8,267)
( 9,268)( 10,269)( 11,270)( 12,271)( 13,272)( 14,273)( 15,274)( 16,275)
( 17,276)( 18,277)( 19,278)( 20,279)( 21,280)( 22,281)( 23,282)( 24,283)
( 25,284)( 26,285)( 27,286)( 28,287)( 29,288)( 30,289)( 31,290)( 32,291)
( 33,292)( 34,293)( 35,294)( 36,295)( 37,296)( 38,223)( 39,224)( 40,225)
( 41,226)( 42,227)( 43,228)( 44,229)( 45,230)( 46,231)( 47,232)( 48,233)
( 49,234)( 50,235)( 51,236)( 52,237)( 53,238)( 54,239)( 55,240)( 56,241)
( 57,242)( 58,243)( 59,244)( 60,245)( 61,246)( 62,247)( 63,248)( 64,249)
( 65,250)( 66,251)( 67,252)( 68,253)( 69,254)( 70,255)( 71,256)( 72,257)
( 73,258)( 74,259)( 75,297)( 76,298)( 77,299)( 78,300)( 79,301)( 80,302)
( 81,303)( 82,304)( 83,305)( 84,306)( 85,307)( 86,308)( 87,309)( 88,310)
( 89,311)( 90,312)( 91,313)( 92,314)( 93,315)( 94,316)( 95,317)( 96,318)
( 97,319)( 98,320)( 99,321)(100,322)(101,323)(102,324)(103,325)(104,326)
(105,327)(106,328)(107,329)(108,330)(109,331)(110,332)(111,333)(112,371)
(113,372)(114,373)(115,374)(116,375)(117,376)(118,377)(119,378)(120,379)
(121,380)(122,381)(123,382)(124,383)(125,384)(126,385)(127,386)(128,387)
(129,388)(130,389)(131,390)(132,391)(133,392)(134,393)(135,394)(136,395)
(137,396)(138,397)(139,398)(140,399)(141,400)(142,401)(143,402)(144,403)
(145,404)(146,405)(147,406)(148,407)(149,334)(150,335)(151,336)(152,337)
(153,338)(154,339)(155,340)(156,341)(157,342)(158,343)(159,344)(160,345)
(161,346)(162,347)(163,348)(164,349)(165,350)(166,351)(167,352)(168,353)
(169,354)(170,355)(171,356)(172,357)(173,358)(174,359)(175,360)(176,361)
(177,362)(178,363)(179,364)(180,365)(181,366)(182,367)(183,368)(184,369)
(185,370)(186,408)(187,409)(188,410)(189,411)(190,412)(191,413)(192,414)
(193,415)(194,416)(195,417)(196,418)(197,419)(198,420)(199,421)(200,422)
(201,423)(202,424)(203,425)(204,426)(205,427)(206,428)(207,429)(208,430)
(209,431)(210,432)(211,433)(212,434)(213,435)(214,436)(215,437)(216,438)
(217,439)(218,440)(219,441)(220,442)(221,443)(222,444);;
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,
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*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*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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(444)!( 2, 37)( 3, 36)( 4, 35)( 5, 34)( 6, 33)( 7, 32)( 8, 31)
( 9, 30)( 10, 29)( 11, 28)( 12, 27)( 13, 26)( 14, 25)( 15, 24)( 16, 23)
( 17, 22)( 18, 21)( 19, 20)( 39, 74)( 40, 73)( 41, 72)( 42, 71)( 43, 70)
( 44, 69)( 45, 68)( 46, 67)( 47, 66)( 48, 65)( 49, 64)( 50, 63)( 51, 62)
( 52, 61)( 53, 60)( 54, 59)( 55, 58)( 56, 57)( 76,111)( 77,110)( 78,109)
( 79,108)( 80,107)( 81,106)( 82,105)( 83,104)( 84,103)( 85,102)( 86,101)
( 87,100)( 88, 99)( 89, 98)( 90, 97)( 91, 96)( 92, 95)( 93, 94)(113,148)
(114,147)(115,146)(116,145)(117,144)(118,143)(119,142)(120,141)(121,140)
(122,139)(123,138)(124,137)(125,136)(126,135)(127,134)(128,133)(129,132)
(130,131)(150,185)(151,184)(152,183)(153,182)(154,181)(155,180)(156,179)
(157,178)(158,177)(159,176)(160,175)(161,174)(162,173)(163,172)(164,171)
(165,170)(166,169)(167,168)(187,222)(188,221)(189,220)(190,219)(191,218)
(192,217)(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(199,210)
(200,209)(201,208)(202,207)(203,206)(204,205)(224,259)(225,258)(226,257)
(227,256)(228,255)(229,254)(230,253)(231,252)(232,251)(233,250)(234,249)
(235,248)(236,247)(237,246)(238,245)(239,244)(240,243)(241,242)(261,296)
(262,295)(263,294)(264,293)(265,292)(266,291)(267,290)(268,289)(269,288)
(270,287)(271,286)(272,285)(273,284)(274,283)(275,282)(276,281)(277,280)
(278,279)(298,333)(299,332)(300,331)(301,330)(302,329)(303,328)(304,327)
(305,326)(306,325)(307,324)(308,323)(309,322)(310,321)(311,320)(312,319)
(313,318)(314,317)(315,316)(335,370)(336,369)(337,368)(338,367)(339,366)
(340,365)(341,364)(342,363)(343,362)(344,361)(345,360)(346,359)(347,358)
(348,357)(349,356)(350,355)(351,354)(352,353)(372,407)(373,406)(374,405)
(375,404)(376,403)(377,402)(378,401)(379,400)(380,399)(381,398)(382,397)
(383,396)(384,395)(385,394)(386,393)(387,392)(388,391)(389,390)(409,444)
(410,443)(411,442)(412,441)(413,440)(414,439)(415,438)(416,437)(417,436)
(418,435)(419,434)(420,433)(421,432)(422,431)(423,430)(424,429)(425,428)
(426,427);
s1 := Sym(444)!( 1, 2)( 3, 37)( 4, 36)( 5, 35)( 6, 34)( 7, 33)( 8, 32)
( 9, 31)( 10, 30)( 11, 29)( 12, 28)( 13, 27)( 14, 26)( 15, 25)( 16, 24)
( 17, 23)( 18, 22)( 19, 21)( 38, 76)( 39, 75)( 40,111)( 41,110)( 42,109)
( 43,108)( 44,107)( 45,106)( 46,105)( 47,104)( 48,103)( 49,102)( 50,101)
( 51,100)( 52, 99)( 53, 98)( 54, 97)( 55, 96)( 56, 95)( 57, 94)( 58, 93)
( 59, 92)( 60, 91)( 61, 90)( 62, 89)( 63, 88)( 64, 87)( 65, 86)( 66, 85)
( 67, 84)( 68, 83)( 69, 82)( 70, 81)( 71, 80)( 72, 79)( 73, 78)( 74, 77)
(112,113)(114,148)(115,147)(116,146)(117,145)(118,144)(119,143)(120,142)
(121,141)(122,140)(123,139)(124,138)(125,137)(126,136)(127,135)(128,134)
(129,133)(130,132)(149,187)(150,186)(151,222)(152,221)(153,220)(154,219)
(155,218)(156,217)(157,216)(158,215)(159,214)(160,213)(161,212)(162,211)
(163,210)(164,209)(165,208)(166,207)(167,206)(168,205)(169,204)(170,203)
(171,202)(172,201)(173,200)(174,199)(175,198)(176,197)(177,196)(178,195)
(179,194)(180,193)(181,192)(182,191)(183,190)(184,189)(185,188)(223,335)
(224,334)(225,370)(226,369)(227,368)(228,367)(229,366)(230,365)(231,364)
(232,363)(233,362)(234,361)(235,360)(236,359)(237,358)(238,357)(239,356)
(240,355)(241,354)(242,353)(243,352)(244,351)(245,350)(246,349)(247,348)
(248,347)(249,346)(250,345)(251,344)(252,343)(253,342)(254,341)(255,340)
(256,339)(257,338)(258,337)(259,336)(260,409)(261,408)(262,444)(263,443)
(264,442)(265,441)(266,440)(267,439)(268,438)(269,437)(270,436)(271,435)
(272,434)(273,433)(274,432)(275,431)(276,430)(277,429)(278,428)(279,427)
(280,426)(281,425)(282,424)(283,423)(284,422)(285,421)(286,420)(287,419)
(288,418)(289,417)(290,416)(291,415)(292,414)(293,413)(294,412)(295,411)
(296,410)(297,372)(298,371)(299,407)(300,406)(301,405)(302,404)(303,403)
(304,402)(305,401)(306,400)(307,399)(308,398)(309,397)(310,396)(311,395)
(312,394)(313,393)(314,392)(315,391)(316,390)(317,389)(318,388)(319,387)
(320,386)(321,385)(322,384)(323,383)(324,382)(325,381)(326,380)(327,379)
(328,378)(329,377)(330,376)(331,375)(332,374)(333,373);
s2 := Sym(444)!( 1,260)( 2,261)( 3,262)( 4,263)( 5,264)( 6,265)( 7,266)
( 8,267)( 9,268)( 10,269)( 11,270)( 12,271)( 13,272)( 14,273)( 15,274)
( 16,275)( 17,276)( 18,277)( 19,278)( 20,279)( 21,280)( 22,281)( 23,282)
( 24,283)( 25,284)( 26,285)( 27,286)( 28,287)( 29,288)( 30,289)( 31,290)
( 32,291)( 33,292)( 34,293)( 35,294)( 36,295)( 37,296)( 38,223)( 39,224)
( 40,225)( 41,226)( 42,227)( 43,228)( 44,229)( 45,230)( 46,231)( 47,232)
( 48,233)( 49,234)( 50,235)( 51,236)( 52,237)( 53,238)( 54,239)( 55,240)
( 56,241)( 57,242)( 58,243)( 59,244)( 60,245)( 61,246)( 62,247)( 63,248)
( 64,249)( 65,250)( 66,251)( 67,252)( 68,253)( 69,254)( 70,255)( 71,256)
( 72,257)( 73,258)( 74,259)( 75,297)( 76,298)( 77,299)( 78,300)( 79,301)
( 80,302)( 81,303)( 82,304)( 83,305)( 84,306)( 85,307)( 86,308)( 87,309)
( 88,310)( 89,311)( 90,312)( 91,313)( 92,314)( 93,315)( 94,316)( 95,317)
( 96,318)( 97,319)( 98,320)( 99,321)(100,322)(101,323)(102,324)(103,325)
(104,326)(105,327)(106,328)(107,329)(108,330)(109,331)(110,332)(111,333)
(112,371)(113,372)(114,373)(115,374)(116,375)(117,376)(118,377)(119,378)
(120,379)(121,380)(122,381)(123,382)(124,383)(125,384)(126,385)(127,386)
(128,387)(129,388)(130,389)(131,390)(132,391)(133,392)(134,393)(135,394)
(136,395)(137,396)(138,397)(139,398)(140,399)(141,400)(142,401)(143,402)
(144,403)(145,404)(146,405)(147,406)(148,407)(149,334)(150,335)(151,336)
(152,337)(153,338)(154,339)(155,340)(156,341)(157,342)(158,343)(159,344)
(160,345)(161,346)(162,347)(163,348)(164,349)(165,350)(166,351)(167,352)
(168,353)(169,354)(170,355)(171,356)(172,357)(173,358)(174,359)(175,360)
(176,361)(177,362)(178,363)(179,364)(180,365)(181,366)(182,367)(183,368)
(184,369)(185,370)(186,408)(187,409)(188,410)(189,411)(190,412)(191,413)
(192,414)(193,415)(194,416)(195,417)(196,418)(197,419)(198,420)(199,421)
(200,422)(201,423)(202,424)(203,425)(204,426)(205,427)(206,428)(207,429)
(208,430)(209,431)(210,432)(211,433)(212,434)(213,435)(214,436)(215,437)
(216,438)(217,439)(218,440)(219,441)(220,442)(221,443)(222,444);
poly := sub<Sym(444)|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,
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*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*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*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 >;
References : None.
to this polytope