Polytope of Type {42,22}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,22}*1848
Also Known As : {42,22|2}. if this polytope has another name.
Group : SmallGroup(1848,147)
Rank : 3
Schlafli Type : {42,22}
Number of vertices, edges, etc : 42, 462, 22
Order of s0s1s2 : 462
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) :
3-fold quotients : {14,22}*616
7-fold quotients : {6,22}*264
11-fold quotients : {42,2}*168
21-fold quotients : {2,22}*88
22-fold quotients : {21,2}*84
33-fold quotients : {14,2}*56
42-fold quotients : {2,11}*44
66-fold quotients : {7,2}*28
77-fold quotients : {6,2}*24
154-fold quotients : {3,2}*12
231-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 12, 67)( 13, 68)( 14, 69)( 15, 70)( 16, 71)( 17, 72)( 18, 73)( 19, 74)
( 20, 75)( 21, 76)( 22, 77)( 23, 56)( 24, 57)( 25, 58)( 26, 59)( 27, 60)
( 28, 61)( 29, 62)( 30, 63)( 31, 64)( 32, 65)( 33, 66)( 34, 45)( 35, 46)
( 36, 47)( 37, 48)( 38, 49)( 39, 50)( 40, 51)( 41, 52)( 42, 53)( 43, 54)
( 44, 55)( 78,155)( 79,156)( 80,157)( 81,158)( 82,159)( 83,160)( 84,161)
( 85,162)( 86,163)( 87,164)( 88,165)( 89,221)( 90,222)( 91,223)( 92,224)
( 93,225)( 94,226)( 95,227)( 96,228)( 97,229)( 98,230)( 99,231)(100,210)
(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)(108,218)
(109,219)(110,220)(111,199)(112,200)(113,201)(114,202)(115,203)(116,204)
(117,205)(118,206)(119,207)(120,208)(121,209)(122,188)(123,189)(124,190)
(125,191)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)(132,198)
(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)(140,184)
(141,185)(142,186)(143,187)(144,166)(145,167)(146,168)(147,169)(148,170)
(149,171)(150,172)(151,173)(152,174)(153,175)(154,176)(243,298)(244,299)
(245,300)(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)(252,307)
(253,308)(254,287)(255,288)(256,289)(257,290)(258,291)(259,292)(260,293)
(261,294)(262,295)(263,296)(264,297)(265,276)(266,277)(267,278)(268,279)
(269,280)(270,281)(271,282)(272,283)(273,284)(274,285)(275,286)(309,386)
(310,387)(311,388)(312,389)(313,390)(314,391)(315,392)(316,393)(317,394)
(318,395)(319,396)(320,452)(321,453)(322,454)(323,455)(324,456)(325,457)
(326,458)(327,459)(328,460)(329,461)(330,462)(331,441)(332,442)(333,443)
(334,444)(335,445)(336,446)(337,447)(338,448)(339,449)(340,450)(341,451)
(342,430)(343,431)(344,432)(345,433)(346,434)(347,435)(348,436)(349,437)
(350,438)(351,439)(352,440)(353,419)(354,420)(355,421)(356,422)(357,423)
(358,424)(359,425)(360,426)(361,427)(362,428)(363,429)(364,408)(365,409)
(366,410)(367,411)(368,412)(369,413)(370,414)(371,415)(372,416)(373,417)
(374,418)(375,397)(376,398)(377,399)(378,400)(379,401)(380,402)(381,403)
(382,404)(383,405)(384,406)(385,407);;
s1 := ( 1, 89)( 2, 99)( 3, 98)( 4, 97)( 5, 96)( 6, 95)( 7, 94)( 8, 93)
( 9, 92)( 10, 91)( 11, 90)( 12, 78)( 13, 88)( 14, 87)( 15, 86)( 16, 85)
( 17, 84)( 18, 83)( 19, 82)( 20, 81)( 21, 80)( 22, 79)( 23,144)( 24,154)
( 25,153)( 26,152)( 27,151)( 28,150)( 29,149)( 30,148)( 31,147)( 32,146)
( 33,145)( 34,133)( 35,143)( 36,142)( 37,141)( 38,140)( 39,139)( 40,138)
( 41,137)( 42,136)( 43,135)( 44,134)( 45,122)( 46,132)( 47,131)( 48,130)
( 49,129)( 50,128)( 51,127)( 52,126)( 53,125)( 54,124)( 55,123)( 56,111)
( 57,121)( 58,120)( 59,119)( 60,118)( 61,117)( 62,116)( 63,115)( 64,114)
( 65,113)( 66,112)( 67,100)( 68,110)( 69,109)( 70,108)( 71,107)( 72,106)
( 73,105)( 74,104)( 75,103)( 76,102)( 77,101)(155,166)(156,176)(157,175)
(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)(165,167)
(177,221)(178,231)(179,230)(180,229)(181,228)(182,227)(183,226)(184,225)
(185,224)(186,223)(187,222)(188,210)(189,220)(190,219)(191,218)(192,217)
(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(200,209)(201,208)
(202,207)(203,206)(204,205)(232,320)(233,330)(234,329)(235,328)(236,327)
(237,326)(238,325)(239,324)(240,323)(241,322)(242,321)(243,309)(244,319)
(245,318)(246,317)(247,316)(248,315)(249,314)(250,313)(251,312)(252,311)
(253,310)(254,375)(255,385)(256,384)(257,383)(258,382)(259,381)(260,380)
(261,379)(262,378)(263,377)(264,376)(265,364)(266,374)(267,373)(268,372)
(269,371)(270,370)(271,369)(272,368)(273,367)(274,366)(275,365)(276,353)
(277,363)(278,362)(279,361)(280,360)(281,359)(282,358)(283,357)(284,356)
(285,355)(286,354)(287,342)(288,352)(289,351)(290,350)(291,349)(292,348)
(293,347)(294,346)(295,345)(296,344)(297,343)(298,331)(299,341)(300,340)
(301,339)(302,338)(303,337)(304,336)(305,335)(306,334)(307,333)(308,332)
(386,397)(387,407)(388,406)(389,405)(390,404)(391,403)(392,402)(393,401)
(394,400)(395,399)(396,398)(408,452)(409,462)(410,461)(411,460)(412,459)
(413,458)(414,457)(415,456)(416,455)(417,454)(418,453)(419,441)(420,451)
(421,450)(422,449)(423,448)(424,447)(425,446)(426,445)(427,444)(428,443)
(429,442)(431,440)(432,439)(433,438)(434,437)(435,436);;
s2 := ( 1,233)( 2,232)( 3,242)( 4,241)( 5,240)( 6,239)( 7,238)( 8,237)
( 9,236)( 10,235)( 11,234)( 12,244)( 13,243)( 14,253)( 15,252)( 16,251)
( 17,250)( 18,249)( 19,248)( 20,247)( 21,246)( 22,245)( 23,255)( 24,254)
( 25,264)( 26,263)( 27,262)( 28,261)( 29,260)( 30,259)( 31,258)( 32,257)
( 33,256)( 34,266)( 35,265)( 36,275)( 37,274)( 38,273)( 39,272)( 40,271)
( 41,270)( 42,269)( 43,268)( 44,267)( 45,277)( 46,276)( 47,286)( 48,285)
( 49,284)( 50,283)( 51,282)( 52,281)( 53,280)( 54,279)( 55,278)( 56,288)
( 57,287)( 58,297)( 59,296)( 60,295)( 61,294)( 62,293)( 63,292)( 64,291)
( 65,290)( 66,289)( 67,299)( 68,298)( 69,308)( 70,307)( 71,306)( 72,305)
( 73,304)( 74,303)( 75,302)( 76,301)( 77,300)( 78,310)( 79,309)( 80,319)
( 81,318)( 82,317)( 83,316)( 84,315)( 85,314)( 86,313)( 87,312)( 88,311)
( 89,321)( 90,320)( 91,330)( 92,329)( 93,328)( 94,327)( 95,326)( 96,325)
( 97,324)( 98,323)( 99,322)(100,332)(101,331)(102,341)(103,340)(104,339)
(105,338)(106,337)(107,336)(108,335)(109,334)(110,333)(111,343)(112,342)
(113,352)(114,351)(115,350)(116,349)(117,348)(118,347)(119,346)(120,345)
(121,344)(122,354)(123,353)(124,363)(125,362)(126,361)(127,360)(128,359)
(129,358)(130,357)(131,356)(132,355)(133,365)(134,364)(135,374)(136,373)
(137,372)(138,371)(139,370)(140,369)(141,368)(142,367)(143,366)(144,376)
(145,375)(146,385)(147,384)(148,383)(149,382)(150,381)(151,380)(152,379)
(153,378)(154,377)(155,387)(156,386)(157,396)(158,395)(159,394)(160,393)
(161,392)(162,391)(163,390)(164,389)(165,388)(166,398)(167,397)(168,407)
(169,406)(170,405)(171,404)(172,403)(173,402)(174,401)(175,400)(176,399)
(177,409)(178,408)(179,418)(180,417)(181,416)(182,415)(183,414)(184,413)
(185,412)(186,411)(187,410)(188,420)(189,419)(190,429)(191,428)(192,427)
(193,426)(194,425)(195,424)(196,423)(197,422)(198,421)(199,431)(200,430)
(201,440)(202,439)(203,438)(204,437)(205,436)(206,435)(207,434)(208,433)
(209,432)(210,442)(211,441)(212,451)(213,450)(214,449)(215,448)(216,447)
(217,446)(218,445)(219,444)(220,443)(221,453)(222,452)(223,462)(224,461)
(225,460)(226,459)(227,458)(228,457)(229,456)(230,455)(231,454);;
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*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(462)!( 12, 67)( 13, 68)( 14, 69)( 15, 70)( 16, 71)( 17, 72)( 18, 73)
( 19, 74)( 20, 75)( 21, 76)( 22, 77)( 23, 56)( 24, 57)( 25, 58)( 26, 59)
( 27, 60)( 28, 61)( 29, 62)( 30, 63)( 31, 64)( 32, 65)( 33, 66)( 34, 45)
( 35, 46)( 36, 47)( 37, 48)( 38, 49)( 39, 50)( 40, 51)( 41, 52)( 42, 53)
( 43, 54)( 44, 55)( 78,155)( 79,156)( 80,157)( 81,158)( 82,159)( 83,160)
( 84,161)( 85,162)( 86,163)( 87,164)( 88,165)( 89,221)( 90,222)( 91,223)
( 92,224)( 93,225)( 94,226)( 95,227)( 96,228)( 97,229)( 98,230)( 99,231)
(100,210)(101,211)(102,212)(103,213)(104,214)(105,215)(106,216)(107,217)
(108,218)(109,219)(110,220)(111,199)(112,200)(113,201)(114,202)(115,203)
(116,204)(117,205)(118,206)(119,207)(120,208)(121,209)(122,188)(123,189)
(124,190)(125,191)(126,192)(127,193)(128,194)(129,195)(130,196)(131,197)
(132,198)(133,177)(134,178)(135,179)(136,180)(137,181)(138,182)(139,183)
(140,184)(141,185)(142,186)(143,187)(144,166)(145,167)(146,168)(147,169)
(148,170)(149,171)(150,172)(151,173)(152,174)(153,175)(154,176)(243,298)
(244,299)(245,300)(246,301)(247,302)(248,303)(249,304)(250,305)(251,306)
(252,307)(253,308)(254,287)(255,288)(256,289)(257,290)(258,291)(259,292)
(260,293)(261,294)(262,295)(263,296)(264,297)(265,276)(266,277)(267,278)
(268,279)(269,280)(270,281)(271,282)(272,283)(273,284)(274,285)(275,286)
(309,386)(310,387)(311,388)(312,389)(313,390)(314,391)(315,392)(316,393)
(317,394)(318,395)(319,396)(320,452)(321,453)(322,454)(323,455)(324,456)
(325,457)(326,458)(327,459)(328,460)(329,461)(330,462)(331,441)(332,442)
(333,443)(334,444)(335,445)(336,446)(337,447)(338,448)(339,449)(340,450)
(341,451)(342,430)(343,431)(344,432)(345,433)(346,434)(347,435)(348,436)
(349,437)(350,438)(351,439)(352,440)(353,419)(354,420)(355,421)(356,422)
(357,423)(358,424)(359,425)(360,426)(361,427)(362,428)(363,429)(364,408)
(365,409)(366,410)(367,411)(368,412)(369,413)(370,414)(371,415)(372,416)
(373,417)(374,418)(375,397)(376,398)(377,399)(378,400)(379,401)(380,402)
(381,403)(382,404)(383,405)(384,406)(385,407);
s1 := Sym(462)!( 1, 89)( 2, 99)( 3, 98)( 4, 97)( 5, 96)( 6, 95)( 7, 94)
( 8, 93)( 9, 92)( 10, 91)( 11, 90)( 12, 78)( 13, 88)( 14, 87)( 15, 86)
( 16, 85)( 17, 84)( 18, 83)( 19, 82)( 20, 81)( 21, 80)( 22, 79)( 23,144)
( 24,154)( 25,153)( 26,152)( 27,151)( 28,150)( 29,149)( 30,148)( 31,147)
( 32,146)( 33,145)( 34,133)( 35,143)( 36,142)( 37,141)( 38,140)( 39,139)
( 40,138)( 41,137)( 42,136)( 43,135)( 44,134)( 45,122)( 46,132)( 47,131)
( 48,130)( 49,129)( 50,128)( 51,127)( 52,126)( 53,125)( 54,124)( 55,123)
( 56,111)( 57,121)( 58,120)( 59,119)( 60,118)( 61,117)( 62,116)( 63,115)
( 64,114)( 65,113)( 66,112)( 67,100)( 68,110)( 69,109)( 70,108)( 71,107)
( 72,106)( 73,105)( 74,104)( 75,103)( 76,102)( 77,101)(155,166)(156,176)
(157,175)(158,174)(159,173)(160,172)(161,171)(162,170)(163,169)(164,168)
(165,167)(177,221)(178,231)(179,230)(180,229)(181,228)(182,227)(183,226)
(184,225)(185,224)(186,223)(187,222)(188,210)(189,220)(190,219)(191,218)
(192,217)(193,216)(194,215)(195,214)(196,213)(197,212)(198,211)(200,209)
(201,208)(202,207)(203,206)(204,205)(232,320)(233,330)(234,329)(235,328)
(236,327)(237,326)(238,325)(239,324)(240,323)(241,322)(242,321)(243,309)
(244,319)(245,318)(246,317)(247,316)(248,315)(249,314)(250,313)(251,312)
(252,311)(253,310)(254,375)(255,385)(256,384)(257,383)(258,382)(259,381)
(260,380)(261,379)(262,378)(263,377)(264,376)(265,364)(266,374)(267,373)
(268,372)(269,371)(270,370)(271,369)(272,368)(273,367)(274,366)(275,365)
(276,353)(277,363)(278,362)(279,361)(280,360)(281,359)(282,358)(283,357)
(284,356)(285,355)(286,354)(287,342)(288,352)(289,351)(290,350)(291,349)
(292,348)(293,347)(294,346)(295,345)(296,344)(297,343)(298,331)(299,341)
(300,340)(301,339)(302,338)(303,337)(304,336)(305,335)(306,334)(307,333)
(308,332)(386,397)(387,407)(388,406)(389,405)(390,404)(391,403)(392,402)
(393,401)(394,400)(395,399)(396,398)(408,452)(409,462)(410,461)(411,460)
(412,459)(413,458)(414,457)(415,456)(416,455)(417,454)(418,453)(419,441)
(420,451)(421,450)(422,449)(423,448)(424,447)(425,446)(426,445)(427,444)
(428,443)(429,442)(431,440)(432,439)(433,438)(434,437)(435,436);
s2 := Sym(462)!( 1,233)( 2,232)( 3,242)( 4,241)( 5,240)( 6,239)( 7,238)
( 8,237)( 9,236)( 10,235)( 11,234)( 12,244)( 13,243)( 14,253)( 15,252)
( 16,251)( 17,250)( 18,249)( 19,248)( 20,247)( 21,246)( 22,245)( 23,255)
( 24,254)( 25,264)( 26,263)( 27,262)( 28,261)( 29,260)( 30,259)( 31,258)
( 32,257)( 33,256)( 34,266)( 35,265)( 36,275)( 37,274)( 38,273)( 39,272)
( 40,271)( 41,270)( 42,269)( 43,268)( 44,267)( 45,277)( 46,276)( 47,286)
( 48,285)( 49,284)( 50,283)( 51,282)( 52,281)( 53,280)( 54,279)( 55,278)
( 56,288)( 57,287)( 58,297)( 59,296)( 60,295)( 61,294)( 62,293)( 63,292)
( 64,291)( 65,290)( 66,289)( 67,299)( 68,298)( 69,308)( 70,307)( 71,306)
( 72,305)( 73,304)( 74,303)( 75,302)( 76,301)( 77,300)( 78,310)( 79,309)
( 80,319)( 81,318)( 82,317)( 83,316)( 84,315)( 85,314)( 86,313)( 87,312)
( 88,311)( 89,321)( 90,320)( 91,330)( 92,329)( 93,328)( 94,327)( 95,326)
( 96,325)( 97,324)( 98,323)( 99,322)(100,332)(101,331)(102,341)(103,340)
(104,339)(105,338)(106,337)(107,336)(108,335)(109,334)(110,333)(111,343)
(112,342)(113,352)(114,351)(115,350)(116,349)(117,348)(118,347)(119,346)
(120,345)(121,344)(122,354)(123,353)(124,363)(125,362)(126,361)(127,360)
(128,359)(129,358)(130,357)(131,356)(132,355)(133,365)(134,364)(135,374)
(136,373)(137,372)(138,371)(139,370)(140,369)(141,368)(142,367)(143,366)
(144,376)(145,375)(146,385)(147,384)(148,383)(149,382)(150,381)(151,380)
(152,379)(153,378)(154,377)(155,387)(156,386)(157,396)(158,395)(159,394)
(160,393)(161,392)(162,391)(163,390)(164,389)(165,388)(166,398)(167,397)
(168,407)(169,406)(170,405)(171,404)(172,403)(173,402)(174,401)(175,400)
(176,399)(177,409)(178,408)(179,418)(180,417)(181,416)(182,415)(183,414)
(184,413)(185,412)(186,411)(187,410)(188,420)(189,419)(190,429)(191,428)
(192,427)(193,426)(194,425)(195,424)(196,423)(197,422)(198,421)(199,431)
(200,430)(201,440)(202,439)(203,438)(204,437)(205,436)(206,435)(207,434)
(208,433)(209,432)(210,442)(211,441)(212,451)(213,450)(214,449)(215,448)
(216,447)(217,446)(218,445)(219,444)(220,443)(221,453)(222,452)(223,462)
(224,461)(225,460)(226,459)(227,458)(228,457)(229,456)(230,455)(231,454);
poly := sub<Sym(462)|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*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 >;
References : None.
to this polytope