Polytope of Type {4,222}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,222}*1776a
Also Known As : {4,222|2}. if this polytope has another name.
Group : SmallGroup(1776,228)
Rank : 3
Schlafli Type : {4,222}
Number of vertices, edges, etc : 4, 444, 222
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 : {2,222}*888
   3-fold quotients : {4,74}*592
   4-fold quotients : {2,111}*444
   6-fold quotients : {2,74}*296
   12-fold quotients : {2,37}*148
   37-fold quotients : {4,6}*48a
   74-fold quotients : {2,6}*24
   111-fold quotients : {4,2}*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 := (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)(260,371)(261,372)(262,373)
(263,374)(264,375)(265,376)(266,377)(267,378)(268,379)(269,380)(270,381)
(271,382)(272,383)(273,384)(274,385)(275,386)(276,387)(277,388)(278,389)
(279,390)(280,391)(281,392)(282,393)(283,394)(284,395)(285,396)(286,397)
(287,398)(288,399)(289,400)(290,401)(291,402)(292,403)(293,404)(294,405)
(295,406)(296,407)(297,408)(298,409)(299,410)(300,411)(301,412)(302,413)
(303,414)(304,415)(305,416)(306,417)(307,418)(308,419)(309,420)(310,421)
(311,422)(312,423)(313,424)(314,425)(315,426)(316,427)(317,428)(318,429)
(319,430)(320,431)(321,432)(322,433)(323,434)(324,435)(325,436)(326,437)
(327,438)(328,439)(329,440)(330,441)(331,442)(332,443)(333,444);;
s1 := (  1,223)(  2,259)(  3,258)(  4,257)(  5,256)(  6,255)(  7,254)(  8,253)
(  9,252)( 10,251)( 11,250)( 12,249)( 13,248)( 14,247)( 15,246)( 16,245)
( 17,244)( 18,243)( 19,242)( 20,241)( 21,240)( 22,239)( 23,238)( 24,237)
( 25,236)( 26,235)( 27,234)( 28,233)( 29,232)( 30,231)( 31,230)( 32,229)
( 33,228)( 34,227)( 35,226)( 36,225)( 37,224)( 38,297)( 39,333)( 40,332)
( 41,331)( 42,330)( 43,329)( 44,328)( 45,327)( 46,326)( 47,325)( 48,324)
( 49,323)( 50,322)( 51,321)( 52,320)( 53,319)( 54,318)( 55,317)( 56,316)
( 57,315)( 58,314)( 59,313)( 60,312)( 61,311)( 62,310)( 63,309)( 64,308)
( 65,307)( 66,306)( 67,305)( 68,304)( 69,303)( 70,302)( 71,301)( 72,300)
( 73,299)( 74,298)( 75,260)( 76,296)( 77,295)( 78,294)( 79,293)( 80,292)
( 81,291)( 82,290)( 83,289)( 84,288)( 85,287)( 86,286)( 87,285)( 88,284)
( 89,283)( 90,282)( 91,281)( 92,280)( 93,279)( 94,278)( 95,277)( 96,276)
( 97,275)( 98,274)( 99,273)(100,272)(101,271)(102,270)(103,269)(104,268)
(105,267)(106,266)(107,265)(108,264)(109,263)(110,262)(111,261)(112,334)
(113,370)(114,369)(115,368)(116,367)(117,366)(118,365)(119,364)(120,363)
(121,362)(122,361)(123,360)(124,359)(125,358)(126,357)(127,356)(128,355)
(129,354)(130,353)(131,352)(132,351)(133,350)(134,349)(135,348)(136,347)
(137,346)(138,345)(139,344)(140,343)(141,342)(142,341)(143,340)(144,339)
(145,338)(146,337)(147,336)(148,335)(149,408)(150,444)(151,443)(152,442)
(153,441)(154,440)(155,439)(156,438)(157,437)(158,436)(159,435)(160,434)
(161,433)(162,432)(163,431)(164,430)(165,429)(166,428)(167,427)(168,426)
(169,425)(170,424)(171,423)(172,422)(173,421)(174,420)(175,419)(176,418)
(177,417)(178,416)(179,415)(180,414)(181,413)(182,412)(183,411)(184,410)
(185,409)(186,371)(187,407)(188,406)(189,405)(190,404)(191,403)(192,402)
(193,401)(194,400)(195,399)(196,398)(197,397)(198,396)(199,395)(200,394)
(201,393)(202,392)(203,391)(204,390)(205,389)(206,388)(207,387)(208,386)
(209,385)(210,384)(211,383)(212,382)(213,381)(214,380)(215,379)(216,378)
(217,377)(218,376)(219,375)(220,374)(221,373)(222,372);;
s2 := (  1, 39)(  2, 38)(  3, 74)(  4, 73)(  5, 72)(  6, 71)(  7, 70)(  8, 69)
(  9, 68)( 10, 67)( 11, 66)( 12, 65)( 13, 64)( 14, 63)( 15, 62)( 16, 61)
( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 56)( 22, 55)( 23, 54)( 24, 53)
( 25, 52)( 26, 51)( 27, 50)( 28, 49)( 29, 48)( 30, 47)( 31, 46)( 32, 45)
( 33, 44)( 34, 43)( 35, 42)( 36, 41)( 37, 40)( 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)(112,150)
(113,149)(114,185)(115,184)(116,183)(117,182)(118,181)(119,180)(120,179)
(121,178)(122,177)(123,176)(124,175)(125,174)(126,173)(127,172)(128,171)
(129,170)(130,169)(131,168)(132,167)(133,166)(134,165)(135,164)(136,163)
(137,162)(138,161)(139,160)(140,159)(141,158)(142,157)(143,156)(144,155)
(145,154)(146,153)(147,152)(148,151)(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)(223,261)(224,260)
(225,296)(226,295)(227,294)(228,293)(229,292)(230,291)(231,290)(232,289)
(233,288)(234,287)(235,286)(236,285)(237,284)(238,283)(239,282)(240,281)
(241,280)(242,279)(243,278)(244,277)(245,276)(246,275)(247,274)(248,273)
(249,272)(250,271)(251,270)(252,269)(253,268)(254,267)(255,266)(256,265)
(257,264)(258,263)(259,262)(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)(334,372)(335,371)(336,407)
(337,406)(338,405)(339,404)(340,403)(341,402)(342,401)(343,400)(344,399)
(345,398)(346,397)(347,396)(348,395)(349,394)(350,393)(351,392)(352,391)
(353,390)(354,389)(355,388)(356,387)(357,386)(358,385)(359,384)(360,383)
(361,382)(362,381)(363,380)(364,379)(365,378)(366,377)(367,376)(368,375)
(369,374)(370,373)(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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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)!(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)(260,371)(261,372)
(262,373)(263,374)(264,375)(265,376)(266,377)(267,378)(268,379)(269,380)
(270,381)(271,382)(272,383)(273,384)(274,385)(275,386)(276,387)(277,388)
(278,389)(279,390)(280,391)(281,392)(282,393)(283,394)(284,395)(285,396)
(286,397)(287,398)(288,399)(289,400)(290,401)(291,402)(292,403)(293,404)
(294,405)(295,406)(296,407)(297,408)(298,409)(299,410)(300,411)(301,412)
(302,413)(303,414)(304,415)(305,416)(306,417)(307,418)(308,419)(309,420)
(310,421)(311,422)(312,423)(313,424)(314,425)(315,426)(316,427)(317,428)
(318,429)(319,430)(320,431)(321,432)(322,433)(323,434)(324,435)(325,436)
(326,437)(327,438)(328,439)(329,440)(330,441)(331,442)(332,443)(333,444);
s1 := Sym(444)!(  1,223)(  2,259)(  3,258)(  4,257)(  5,256)(  6,255)(  7,254)
(  8,253)(  9,252)( 10,251)( 11,250)( 12,249)( 13,248)( 14,247)( 15,246)
( 16,245)( 17,244)( 18,243)( 19,242)( 20,241)( 21,240)( 22,239)( 23,238)
( 24,237)( 25,236)( 26,235)( 27,234)( 28,233)( 29,232)( 30,231)( 31,230)
( 32,229)( 33,228)( 34,227)( 35,226)( 36,225)( 37,224)( 38,297)( 39,333)
( 40,332)( 41,331)( 42,330)( 43,329)( 44,328)( 45,327)( 46,326)( 47,325)
( 48,324)( 49,323)( 50,322)( 51,321)( 52,320)( 53,319)( 54,318)( 55,317)
( 56,316)( 57,315)( 58,314)( 59,313)( 60,312)( 61,311)( 62,310)( 63,309)
( 64,308)( 65,307)( 66,306)( 67,305)( 68,304)( 69,303)( 70,302)( 71,301)
( 72,300)( 73,299)( 74,298)( 75,260)( 76,296)( 77,295)( 78,294)( 79,293)
( 80,292)( 81,291)( 82,290)( 83,289)( 84,288)( 85,287)( 86,286)( 87,285)
( 88,284)( 89,283)( 90,282)( 91,281)( 92,280)( 93,279)( 94,278)( 95,277)
( 96,276)( 97,275)( 98,274)( 99,273)(100,272)(101,271)(102,270)(103,269)
(104,268)(105,267)(106,266)(107,265)(108,264)(109,263)(110,262)(111,261)
(112,334)(113,370)(114,369)(115,368)(116,367)(117,366)(118,365)(119,364)
(120,363)(121,362)(122,361)(123,360)(124,359)(125,358)(126,357)(127,356)
(128,355)(129,354)(130,353)(131,352)(132,351)(133,350)(134,349)(135,348)
(136,347)(137,346)(138,345)(139,344)(140,343)(141,342)(142,341)(143,340)
(144,339)(145,338)(146,337)(147,336)(148,335)(149,408)(150,444)(151,443)
(152,442)(153,441)(154,440)(155,439)(156,438)(157,437)(158,436)(159,435)
(160,434)(161,433)(162,432)(163,431)(164,430)(165,429)(166,428)(167,427)
(168,426)(169,425)(170,424)(171,423)(172,422)(173,421)(174,420)(175,419)
(176,418)(177,417)(178,416)(179,415)(180,414)(181,413)(182,412)(183,411)
(184,410)(185,409)(186,371)(187,407)(188,406)(189,405)(190,404)(191,403)
(192,402)(193,401)(194,400)(195,399)(196,398)(197,397)(198,396)(199,395)
(200,394)(201,393)(202,392)(203,391)(204,390)(205,389)(206,388)(207,387)
(208,386)(209,385)(210,384)(211,383)(212,382)(213,381)(214,380)(215,379)
(216,378)(217,377)(218,376)(219,375)(220,374)(221,373)(222,372);
s2 := Sym(444)!(  1, 39)(  2, 38)(  3, 74)(  4, 73)(  5, 72)(  6, 71)(  7, 70)
(  8, 69)(  9, 68)( 10, 67)( 11, 66)( 12, 65)( 13, 64)( 14, 63)( 15, 62)
( 16, 61)( 17, 60)( 18, 59)( 19, 58)( 20, 57)( 21, 56)( 22, 55)( 23, 54)
( 24, 53)( 25, 52)( 26, 51)( 27, 50)( 28, 49)( 29, 48)( 30, 47)( 31, 46)
( 32, 45)( 33, 44)( 34, 43)( 35, 42)( 36, 41)( 37, 40)( 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)
(112,150)(113,149)(114,185)(115,184)(116,183)(117,182)(118,181)(119,180)
(120,179)(121,178)(122,177)(123,176)(124,175)(125,174)(126,173)(127,172)
(128,171)(129,170)(130,169)(131,168)(132,167)(133,166)(134,165)(135,164)
(136,163)(137,162)(138,161)(139,160)(140,159)(141,158)(142,157)(143,156)
(144,155)(145,154)(146,153)(147,152)(148,151)(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)(223,261)
(224,260)(225,296)(226,295)(227,294)(228,293)(229,292)(230,291)(231,290)
(232,289)(233,288)(234,287)(235,286)(236,285)(237,284)(238,283)(239,282)
(240,281)(241,280)(242,279)(243,278)(244,277)(245,276)(246,275)(247,274)
(248,273)(249,272)(250,271)(251,270)(252,269)(253,268)(254,267)(255,266)
(256,265)(257,264)(258,263)(259,262)(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)(334,372)(335,371)
(336,407)(337,406)(338,405)(339,404)(340,403)(341,402)(342,401)(343,400)
(344,399)(345,398)(346,397)(347,396)(348,395)(349,394)(350,393)(351,392)
(352,391)(353,390)(354,389)(355,388)(356,387)(357,386)(358,385)(359,384)
(360,383)(361,382)(362,381)(363,380)(364,379)(365,378)(366,377)(367,376)
(368,375)(369,374)(370,373)(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*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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