Polytope of Type {4,198}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,198}*1584a
Also Known As : {4,198|2}. if this polytope has another name.
Group : SmallGroup(1584,156)
Rank : 3
Schlafli Type : {4,198}
Number of vertices, edges, etc : 4, 396, 198
Order of s₀s₁s₂ : 396
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 : {2,198}*792
   3-fold quotients : {4,66}*528a
   4-fold quotients : {2,99}*396
   6-fold quotients : {2,66}*264
   9-fold quotients : {4,22}*176
   11-fold quotients : {4,18}*144a
   12-fold quotients : {2,33}*132
   18-fold quotients : {2,22}*88
   22-fold quotients : {2,18}*72
   33-fold quotients : {4,6}*48a
   36-fold quotients : {2,11}*44
   44-fold quotients : {2,9}*36
   66-fold quotients : {2,6}*24
   99-fold quotients : {4,2}*16
   132-fold quotients : {2,3}*12
   198-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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