Polytope of Type {12,66}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,66}*1584b
Also Known As : {12,66|2}. if this polytope has another name.
Group : SmallGroup(1584,561)
Rank : 3
Schlafli Type : {12,66}
Number of vertices, edges, etc : 12, 396, 66
Order of s₀s₁s₂ : 132
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 : {6,66}*792b
   3-fold quotients : {12,22}*528, {4,66}*528a
   6-fold quotients : {6,22}*264, {2,66}*264
   9-fold quotients : {4,22}*176
   11-fold quotients : {12,6}*144a
   12-fold quotients : {2,33}*132
   18-fold quotients : {2,22}*88
   22-fold quotients : {6,6}*72a
   33-fold quotients : {12,2}*48, {4,6}*48a
   36-fold quotients : {2,11}*44
   66-fold quotients : {2,6}*24, {6,2}*24
   99-fold quotients : {4,2}*16
   132-fold quotients : {2,3}*12, {3,2}*12
   198-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := ( 34, 67)( 35, 68)( 36, 69)( 37, 70)( 38, 71)( 39, 72)( 40, 73)( 41, 74)
( 42, 75)( 43, 76)( 44, 77)( 45, 78)( 46, 79)( 47, 80)( 48, 81)( 49, 82)
( 50, 83)( 51, 84)( 52, 85)( 53, 86)( 54, 87)( 55, 88)( 56, 89)( 57, 90)
( 58, 91)( 59, 92)( 60, 93)( 61, 94)( 62, 95)( 63, 96)( 64, 97)( 65, 98)
( 66, 99)(133,166)(134,167)(135,168)(136,169)(137,170)(138,171)(139,172)
(140,173)(141,174)(142,175)(143,176)(144,177)(145,178)(146,179)(147,180)
(148,181)(149,182)(150,183)(151,184)(152,185)(153,186)(154,187)(155,188)
(156,189)(157,190)(158,191)(159,192)(160,193)(161,194)(162,195)(163,196)
(164,197)(165,198)(199,298)(200,299)(201,300)(202,301)(203,302)(204,303)
(205,304)(206,305)(207,306)(208,307)(209,308)(210,309)(211,310)(212,311)
(213,312)(214,313)(215,314)(216,315)(217,316)(218,317)(219,318)(220,319)
(221,320)(222,321)(223,322)(224,323)(225,324)(226,325)(227,326)(228,327)
(229,328)(230,329)(231,330)(232,364)(233,365)(234,366)(235,367)(236,368)
(237,369)(238,370)(239,371)(240,372)(241,373)(242,374)(243,375)(244,376)
(245,377)(246,378)(247,379)(248,380)(249,381)(250,382)(251,383)(252,384)
(253,385)(254,386)(255,387)(256,388)(257,389)(258,390)(259,391)(260,392)
(261,393)(262,394)(263,395)(264,396)(265,331)(266,332)(267,333)(268,334)
(269,335)(270,336)(271,337)(272,338)(273,339)(274,340)(275,341)(276,342)
(277,343)(278,344)(279,345)(280,346)(281,347)(282,348)(283,349)(284,350)
(285,351)(286,352)(287,353)(288,354)(289,355)(290,356)(291,357)(292,358)
(293,359)(294,360)(295,361)(296,362)(297,363);;
s1 := (  1,232)(  2,242)(  3,241)(  4,240)(  5,239)(  6,238)(  7,237)(  8,236)
(  9,235)( 10,234)( 11,233)( 12,254)( 13,264)( 14,263)( 15,262)( 16,261)
( 17,260)( 18,259)( 19,258)( 20,257)( 21,256)( 22,255)( 23,243)( 24,253)
( 25,252)( 26,251)( 27,250)( 28,249)( 29,248)( 30,247)( 31,246)( 32,245)
( 33,244)( 34,199)( 35,209)( 36,208)( 37,207)( 38,206)( 39,205)( 40,204)
( 41,203)( 42,202)( 43,201)( 44,200)( 45,221)( 46,231)( 47,230)( 48,229)
( 49,228)( 50,227)( 51,226)( 52,225)( 53,224)( 54,223)( 55,222)( 56,210)
( 57,220)( 58,219)( 59,218)( 60,217)( 61,216)( 62,215)( 63,214)( 64,213)
( 65,212)( 66,211)( 67,265)( 68,275)( 69,274)( 70,273)( 71,272)( 72,271)
( 73,270)( 74,269)( 75,268)( 76,267)( 77,266)( 78,287)( 79,297)( 80,296)
( 81,295)( 82,294)( 83,293)( 84,292)( 85,291)( 86,290)( 87,289)( 88,288)
( 89,276)( 90,286)( 91,285)( 92,284)( 93,283)( 94,282)( 95,281)( 96,280)
( 97,279)( 98,278)( 99,277)(100,331)(101,341)(102,340)(103,339)(104,338)
(105,337)(106,336)(107,335)(108,334)(109,333)(110,332)(111,353)(112,363)
(113,362)(114,361)(115,360)(116,359)(117,358)(118,357)(119,356)(120,355)
(121,354)(122,342)(123,352)(124,351)(125,350)(126,349)(127,348)(128,347)
(129,346)(130,345)(131,344)(132,343)(133,298)(134,308)(135,307)(136,306)
(137,305)(138,304)(139,303)(140,302)(141,301)(142,300)(143,299)(144,320)
(145,330)(146,329)(147,328)(148,327)(149,326)(150,325)(151,324)(152,323)
(153,322)(154,321)(155,309)(156,319)(157,318)(158,317)(159,316)(160,315)
(161,314)(162,313)(163,312)(164,311)(165,310)(166,364)(167,374)(168,373)
(169,372)(170,371)(171,370)(172,369)(173,368)(174,367)(175,366)(176,365)
(177,386)(178,396)(179,395)(180,394)(181,393)(182,392)(183,391)(184,390)
(185,389)(186,388)(187,387)(188,375)(189,385)(190,384)(191,383)(192,382)
(193,381)(194,380)(195,379)(196,378)(197,377)(198,376);;
s2 := (  1, 13)(  2, 12)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)(  8, 17)
(  9, 16)( 10, 15)( 11, 14)( 23, 24)( 25, 33)( 26, 32)( 27, 31)( 28, 30)
( 34, 46)( 35, 45)( 36, 55)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)
( 42, 49)( 43, 48)( 44, 47)( 56, 57)( 58, 66)( 59, 65)( 60, 64)( 61, 63)
( 67, 79)( 68, 78)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73, 84)( 74, 83)
( 75, 82)( 76, 81)( 77, 80)( 89, 90)( 91, 99)( 92, 98)( 93, 97)( 94, 96)
(100,112)(101,111)(102,121)(103,120)(104,119)(105,118)(106,117)(107,116)
(108,115)(109,114)(110,113)(122,123)(124,132)(125,131)(126,130)(127,129)
(133,145)(134,144)(135,154)(136,153)(137,152)(138,151)(139,150)(140,149)
(141,148)(142,147)(143,146)(155,156)(157,165)(158,164)(159,163)(160,162)
(166,178)(167,177)(168,187)(169,186)(170,185)(171,184)(172,183)(173,182)
(174,181)(175,180)(176,179)(188,189)(190,198)(191,197)(192,196)(193,195)
(199,211)(200,210)(201,220)(202,219)(203,218)(204,217)(205,216)(206,215)
(207,214)(208,213)(209,212)(221,222)(223,231)(224,230)(225,229)(226,228)
(232,244)(233,243)(234,253)(235,252)(236,251)(237,250)(238,249)(239,248)
(240,247)(241,246)(242,245)(254,255)(256,264)(257,263)(258,262)(259,261)
(265,277)(266,276)(267,286)(268,285)(269,284)(270,283)(271,282)(272,281)
(273,280)(274,279)(275,278)(287,288)(289,297)(290,296)(291,295)(292,294)
(298,310)(299,309)(300,319)(301,318)(302,317)(303,316)(304,315)(305,314)
(306,313)(307,312)(308,311)(320,321)(322,330)(323,329)(324,328)(325,327)
(331,343)(332,342)(333,352)(334,351)(335,350)(336,349)(337,348)(338,347)
(339,346)(340,345)(341,344)(353,354)(355,363)(356,362)(357,361)(358,360)
(364,376)(365,375)(366,385)(367,384)(368,383)(369,382)(370,381)(371,380)
(372,379)(373,378)(374,377)(386,387)(388,396)(389,395)(390,394)(391,393);;
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 ];;
poly := F / rels;;

Permutation Representation (Magma) :

s0 := Sym(396)!( 34, 67)( 35, 68)( 36, 69)( 37, 70)( 38, 71)( 39, 72)( 40, 73)
( 41, 74)( 42, 75)( 43, 76)( 44, 77)( 45, 78)( 46, 79)( 47, 80)( 48, 81)
( 49, 82)( 50, 83)( 51, 84)( 52, 85)( 53, 86)( 54, 87)( 55, 88)( 56, 89)
( 57, 90)( 58, 91)( 59, 92)( 60, 93)( 61, 94)( 62, 95)( 63, 96)( 64, 97)
( 65, 98)( 66, 99)(133,166)(134,167)(135,168)(136,169)(137,170)(138,171)
(139,172)(140,173)(141,174)(142,175)(143,176)(144,177)(145,178)(146,179)
(147,180)(148,181)(149,182)(150,183)(151,184)(152,185)(153,186)(154,187)
(155,188)(156,189)(157,190)(158,191)(159,192)(160,193)(161,194)(162,195)
(163,196)(164,197)(165,198)(199,298)(200,299)(201,300)(202,301)(203,302)
(204,303)(205,304)(206,305)(207,306)(208,307)(209,308)(210,309)(211,310)
(212,311)(213,312)(214,313)(215,314)(216,315)(217,316)(218,317)(219,318)
(220,319)(221,320)(222,321)(223,322)(224,323)(225,324)(226,325)(227,326)
(228,327)(229,328)(230,329)(231,330)(232,364)(233,365)(234,366)(235,367)
(236,368)(237,369)(238,370)(239,371)(240,372)(241,373)(242,374)(243,375)
(244,376)(245,377)(246,378)(247,379)(248,380)(249,381)(250,382)(251,383)
(252,384)(253,385)(254,386)(255,387)(256,388)(257,389)(258,390)(259,391)
(260,392)(261,393)(262,394)(263,395)(264,396)(265,331)(266,332)(267,333)
(268,334)(269,335)(270,336)(271,337)(272,338)(273,339)(274,340)(275,341)
(276,342)(277,343)(278,344)(279,345)(280,346)(281,347)(282,348)(283,349)
(284,350)(285,351)(286,352)(287,353)(288,354)(289,355)(290,356)(291,357)
(292,358)(293,359)(294,360)(295,361)(296,362)(297,363);
s1 := Sym(396)!(  1,232)(  2,242)(  3,241)(  4,240)(  5,239)(  6,238)(  7,237)
(  8,236)(  9,235)( 10,234)( 11,233)( 12,254)( 13,264)( 14,263)( 15,262)
( 16,261)( 17,260)( 18,259)( 19,258)( 20,257)( 21,256)( 22,255)( 23,243)
( 24,253)( 25,252)( 26,251)( 27,250)( 28,249)( 29,248)( 30,247)( 31,246)
( 32,245)( 33,244)( 34,199)( 35,209)( 36,208)( 37,207)( 38,206)( 39,205)
( 40,204)( 41,203)( 42,202)( 43,201)( 44,200)( 45,221)( 46,231)( 47,230)
( 48,229)( 49,228)( 50,227)( 51,226)( 52,225)( 53,224)( 54,223)( 55,222)
( 56,210)( 57,220)( 58,219)( 59,218)( 60,217)( 61,216)( 62,215)( 63,214)
( 64,213)( 65,212)( 66,211)( 67,265)( 68,275)( 69,274)( 70,273)( 71,272)
( 72,271)( 73,270)( 74,269)( 75,268)( 76,267)( 77,266)( 78,287)( 79,297)
( 80,296)( 81,295)( 82,294)( 83,293)( 84,292)( 85,291)( 86,290)( 87,289)
( 88,288)( 89,276)( 90,286)( 91,285)( 92,284)( 93,283)( 94,282)( 95,281)
( 96,280)( 97,279)( 98,278)( 99,277)(100,331)(101,341)(102,340)(103,339)
(104,338)(105,337)(106,336)(107,335)(108,334)(109,333)(110,332)(111,353)
(112,363)(113,362)(114,361)(115,360)(116,359)(117,358)(118,357)(119,356)
(120,355)(121,354)(122,342)(123,352)(124,351)(125,350)(126,349)(127,348)
(128,347)(129,346)(130,345)(131,344)(132,343)(133,298)(134,308)(135,307)
(136,306)(137,305)(138,304)(139,303)(140,302)(141,301)(142,300)(143,299)
(144,320)(145,330)(146,329)(147,328)(148,327)(149,326)(150,325)(151,324)
(152,323)(153,322)(154,321)(155,309)(156,319)(157,318)(158,317)(159,316)
(160,315)(161,314)(162,313)(163,312)(164,311)(165,310)(166,364)(167,374)
(168,373)(169,372)(170,371)(171,370)(172,369)(173,368)(174,367)(175,366)
(176,365)(177,386)(178,396)(179,395)(180,394)(181,393)(182,392)(183,391)
(184,390)(185,389)(186,388)(187,387)(188,375)(189,385)(190,384)(191,383)
(192,382)(193,381)(194,380)(195,379)(196,378)(197,377)(198,376);
s2 := Sym(396)!(  1, 13)(  2, 12)(  3, 22)(  4, 21)(  5, 20)(  6, 19)(  7, 18)
(  8, 17)(  9, 16)( 10, 15)( 11, 14)( 23, 24)( 25, 33)( 26, 32)( 27, 31)
( 28, 30)( 34, 46)( 35, 45)( 36, 55)( 37, 54)( 38, 53)( 39, 52)( 40, 51)
( 41, 50)( 42, 49)( 43, 48)( 44, 47)( 56, 57)( 58, 66)( 59, 65)( 60, 64)
( 61, 63)( 67, 79)( 68, 78)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73, 84)
( 74, 83)( 75, 82)( 76, 81)( 77, 80)( 89, 90)( 91, 99)( 92, 98)( 93, 97)
( 94, 96)(100,112)(101,111)(102,121)(103,120)(104,119)(105,118)(106,117)
(107,116)(108,115)(109,114)(110,113)(122,123)(124,132)(125,131)(126,130)
(127,129)(133,145)(134,144)(135,154)(136,153)(137,152)(138,151)(139,150)
(140,149)(141,148)(142,147)(143,146)(155,156)(157,165)(158,164)(159,163)
(160,162)(166,178)(167,177)(168,187)(169,186)(170,185)(171,184)(172,183)
(173,182)(174,181)(175,180)(176,179)(188,189)(190,198)(191,197)(192,196)
(193,195)(199,211)(200,210)(201,220)(202,219)(203,218)(204,217)(205,216)
(206,215)(207,214)(208,213)(209,212)(221,222)(223,231)(224,230)(225,229)
(226,228)(232,244)(233,243)(234,253)(235,252)(236,251)(237,250)(238,249)
(239,248)(240,247)(241,246)(242,245)(254,255)(256,264)(257,263)(258,262)
(259,261)(265,277)(266,276)(267,286)(268,285)(269,284)(270,283)(271,282)
(272,281)(273,280)(274,279)(275,278)(287,288)(289,297)(290,296)(291,295)
(292,294)(298,310)(299,309)(300,319)(301,318)(302,317)(303,316)(304,315)
(305,314)(306,313)(307,312)(308,311)(320,321)(322,330)(323,329)(324,328)
(325,327)(331,343)(332,342)(333,352)(334,351)(335,350)(336,349)(337,348)
(338,347)(339,346)(340,345)(341,344)(353,354)(355,363)(356,362)(357,361)
(358,360)(364,376)(365,375)(366,385)(367,384)(368,383)(369,382)(370,381)
(371,380)(372,379)(373,378)(374,377)(386,387)(388,396)(389,395)(390,394)
(391,393);
poly := sub<Sym(396)|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 >;

References : None.
to this polytope