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