Polytope of Type {42,21}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {42,21}*1764
if this polytope has a name.
Group : SmallGroup(1764,211)
Rank : 3
Schlafli Type : {42,21}
Number of vertices, edges, etc : 42, 441, 21
Order of s0s1s2 : 42
Order of s0s1s2s1 : 42
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {14,21}*588
7-fold quotients : {6,21}*252
9-fold quotients : {14,7}*196
21-fold quotients : {2,21}*84
49-fold quotients : {6,3}*36
63-fold quotients : {2,7}*28
147-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 7)( 3, 6)( 4, 5)( 9, 14)( 10, 13)( 11, 12)( 16, 21)( 17, 20)
( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)( 37, 42)
( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 50, 99)( 51,105)( 52,104)
( 53,103)( 54,102)( 55,101)( 56,100)( 57,106)( 58,112)( 59,111)( 60,110)
( 61,109)( 62,108)( 63,107)( 64,113)( 65,119)( 66,118)( 67,117)( 68,116)
( 69,115)( 70,114)( 71,120)( 72,126)( 73,125)( 74,124)( 75,123)( 76,122)
( 77,121)( 78,127)( 79,133)( 80,132)( 81,131)( 82,130)( 83,129)( 84,128)
( 85,134)( 86,140)( 87,139)( 88,138)( 89,137)( 90,136)( 91,135)( 92,141)
( 93,147)( 94,146)( 95,145)( 96,144)( 97,143)( 98,142)(149,154)(150,153)
(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)(165,166)(170,175)
(171,174)(172,173)(177,182)(178,181)(179,180)(184,189)(185,188)(186,187)
(191,196)(192,195)(193,194)(197,246)(198,252)(199,251)(200,250)(201,249)
(202,248)(203,247)(204,253)(205,259)(206,258)(207,257)(208,256)(209,255)
(210,254)(211,260)(212,266)(213,265)(214,264)(215,263)(216,262)(217,261)
(218,267)(219,273)(220,272)(221,271)(222,270)(223,269)(224,268)(225,274)
(226,280)(227,279)(228,278)(229,277)(230,276)(231,275)(232,281)(233,287)
(234,286)(235,285)(236,284)(237,283)(238,282)(239,288)(240,294)(241,293)
(242,292)(243,291)(244,290)(245,289)(296,301)(297,300)(298,299)(303,308)
(304,307)(305,306)(310,315)(311,314)(312,313)(317,322)(318,321)(319,320)
(324,329)(325,328)(326,327)(331,336)(332,335)(333,334)(338,343)(339,342)
(340,341)(344,393)(345,399)(346,398)(347,397)(348,396)(349,395)(350,394)
(351,400)(352,406)(353,405)(354,404)(355,403)(356,402)(357,401)(358,407)
(359,413)(360,412)(361,411)(362,410)(363,409)(364,408)(365,414)(366,420)
(367,419)(368,418)(369,417)(370,416)(371,415)(372,421)(373,427)(374,426)
(375,425)(376,424)(377,423)(378,422)(379,428)(380,434)(381,433)(382,432)
(383,431)(384,430)(385,429)(386,435)(387,441)(388,440)(389,439)(390,438)
(391,437)(392,436);;
s1 := ( 1, 51)( 2, 50)( 3, 56)( 4, 55)( 5, 54)( 6, 53)( 7, 52)( 8, 93)
( 9, 92)( 10, 98)( 11, 97)( 12, 96)( 13, 95)( 14, 94)( 15, 86)( 16, 85)
( 17, 91)( 18, 90)( 19, 89)( 20, 88)( 21, 87)( 22, 79)( 23, 78)( 24, 84)
( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 72)( 30, 71)( 31, 77)( 32, 76)
( 33, 75)( 34, 74)( 35, 73)( 36, 65)( 37, 64)( 38, 70)( 39, 69)( 40, 68)
( 41, 67)( 42, 66)( 43, 58)( 44, 57)( 45, 63)( 46, 62)( 47, 61)( 48, 60)
( 49, 59)( 99,100)(101,105)(102,104)(106,142)(107,141)(108,147)(109,146)
(110,145)(111,144)(112,143)(113,135)(114,134)(115,140)(116,139)(117,138)
(118,137)(119,136)(120,128)(121,127)(122,133)(123,132)(124,131)(125,130)
(126,129)(148,345)(149,344)(150,350)(151,349)(152,348)(153,347)(154,346)
(155,387)(156,386)(157,392)(158,391)(159,390)(160,389)(161,388)(162,380)
(163,379)(164,385)(165,384)(166,383)(167,382)(168,381)(169,373)(170,372)
(171,378)(172,377)(173,376)(174,375)(175,374)(176,366)(177,365)(178,371)
(179,370)(180,369)(181,368)(182,367)(183,359)(184,358)(185,364)(186,363)
(187,362)(188,361)(189,360)(190,352)(191,351)(192,357)(193,356)(194,355)
(195,354)(196,353)(197,296)(198,295)(199,301)(200,300)(201,299)(202,298)
(203,297)(204,338)(205,337)(206,343)(207,342)(208,341)(209,340)(210,339)
(211,331)(212,330)(213,336)(214,335)(215,334)(216,333)(217,332)(218,324)
(219,323)(220,329)(221,328)(222,327)(223,326)(224,325)(225,317)(226,316)
(227,322)(228,321)(229,320)(230,319)(231,318)(232,310)(233,309)(234,315)
(235,314)(236,313)(237,312)(238,311)(239,303)(240,302)(241,308)(242,307)
(243,306)(244,305)(245,304)(246,394)(247,393)(248,399)(249,398)(250,397)
(251,396)(252,395)(253,436)(254,435)(255,441)(256,440)(257,439)(258,438)
(259,437)(260,429)(261,428)(262,434)(263,433)(264,432)(265,431)(266,430)
(267,422)(268,421)(269,427)(270,426)(271,425)(272,424)(273,423)(274,415)
(275,414)(276,420)(277,419)(278,418)(279,417)(280,416)(281,408)(282,407)
(283,413)(284,412)(285,411)(286,410)(287,409)(288,401)(289,400)(290,406)
(291,405)(292,404)(293,403)(294,402);;
s2 := ( 1,155)( 2,161)( 3,160)( 4,159)( 5,158)( 6,157)( 7,156)( 8,148)
( 9,154)( 10,153)( 11,152)( 12,151)( 13,150)( 14,149)( 15,190)( 16,196)
( 17,195)( 18,194)( 19,193)( 20,192)( 21,191)( 22,183)( 23,189)( 24,188)
( 25,187)( 26,186)( 27,185)( 28,184)( 29,176)( 30,182)( 31,181)( 32,180)
( 33,179)( 34,178)( 35,177)( 36,169)( 37,175)( 38,174)( 39,173)( 40,172)
( 41,171)( 42,170)( 43,162)( 44,168)( 45,167)( 46,166)( 47,165)( 48,164)
( 49,163)( 50,253)( 51,259)( 52,258)( 53,257)( 54,256)( 55,255)( 56,254)
( 57,246)( 58,252)( 59,251)( 60,250)( 61,249)( 62,248)( 63,247)( 64,288)
( 65,294)( 66,293)( 67,292)( 68,291)( 69,290)( 70,289)( 71,281)( 72,287)
( 73,286)( 74,285)( 75,284)( 76,283)( 77,282)( 78,274)( 79,280)( 80,279)
( 81,278)( 82,277)( 83,276)( 84,275)( 85,267)( 86,273)( 87,272)( 88,271)
( 89,270)( 90,269)( 91,268)( 92,260)( 93,266)( 94,265)( 95,264)( 96,263)
( 97,262)( 98,261)( 99,204)(100,210)(101,209)(102,208)(103,207)(104,206)
(105,205)(106,197)(107,203)(108,202)(109,201)(110,200)(111,199)(112,198)
(113,239)(114,245)(115,244)(116,243)(117,242)(118,241)(119,240)(120,232)
(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,225)(128,231)
(129,230)(130,229)(131,228)(132,227)(133,226)(134,218)(135,224)(136,223)
(137,222)(138,221)(139,220)(140,219)(141,211)(142,217)(143,216)(144,215)
(145,214)(146,213)(147,212)(295,302)(296,308)(297,307)(298,306)(299,305)
(300,304)(301,303)(309,337)(310,343)(311,342)(312,341)(313,340)(314,339)
(315,338)(316,330)(317,336)(318,335)(319,334)(320,333)(321,332)(322,331)
(324,329)(325,328)(326,327)(344,400)(345,406)(346,405)(347,404)(348,403)
(349,402)(350,401)(351,393)(352,399)(353,398)(354,397)(355,396)(356,395)
(357,394)(358,435)(359,441)(360,440)(361,439)(362,438)(363,437)(364,436)
(365,428)(366,434)(367,433)(368,432)(369,431)(370,430)(371,429)(372,421)
(373,427)(374,426)(375,425)(376,424)(377,423)(378,422)(379,414)(380,420)
(381,419)(382,418)(383,417)(384,416)(385,415)(386,407)(387,413)(388,412)
(389,411)(390,410)(391,409)(392,408);;
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, s2*s0*s1*s0*s1*s2*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(441)!( 2, 7)( 3, 6)( 4, 5)( 9, 14)( 10, 13)( 11, 12)( 16, 21)
( 17, 20)( 18, 19)( 23, 28)( 24, 27)( 25, 26)( 30, 35)( 31, 34)( 32, 33)
( 37, 42)( 38, 41)( 39, 40)( 44, 49)( 45, 48)( 46, 47)( 50, 99)( 51,105)
( 52,104)( 53,103)( 54,102)( 55,101)( 56,100)( 57,106)( 58,112)( 59,111)
( 60,110)( 61,109)( 62,108)( 63,107)( 64,113)( 65,119)( 66,118)( 67,117)
( 68,116)( 69,115)( 70,114)( 71,120)( 72,126)( 73,125)( 74,124)( 75,123)
( 76,122)( 77,121)( 78,127)( 79,133)( 80,132)( 81,131)( 82,130)( 83,129)
( 84,128)( 85,134)( 86,140)( 87,139)( 88,138)( 89,137)( 90,136)( 91,135)
( 92,141)( 93,147)( 94,146)( 95,145)( 96,144)( 97,143)( 98,142)(149,154)
(150,153)(151,152)(156,161)(157,160)(158,159)(163,168)(164,167)(165,166)
(170,175)(171,174)(172,173)(177,182)(178,181)(179,180)(184,189)(185,188)
(186,187)(191,196)(192,195)(193,194)(197,246)(198,252)(199,251)(200,250)
(201,249)(202,248)(203,247)(204,253)(205,259)(206,258)(207,257)(208,256)
(209,255)(210,254)(211,260)(212,266)(213,265)(214,264)(215,263)(216,262)
(217,261)(218,267)(219,273)(220,272)(221,271)(222,270)(223,269)(224,268)
(225,274)(226,280)(227,279)(228,278)(229,277)(230,276)(231,275)(232,281)
(233,287)(234,286)(235,285)(236,284)(237,283)(238,282)(239,288)(240,294)
(241,293)(242,292)(243,291)(244,290)(245,289)(296,301)(297,300)(298,299)
(303,308)(304,307)(305,306)(310,315)(311,314)(312,313)(317,322)(318,321)
(319,320)(324,329)(325,328)(326,327)(331,336)(332,335)(333,334)(338,343)
(339,342)(340,341)(344,393)(345,399)(346,398)(347,397)(348,396)(349,395)
(350,394)(351,400)(352,406)(353,405)(354,404)(355,403)(356,402)(357,401)
(358,407)(359,413)(360,412)(361,411)(362,410)(363,409)(364,408)(365,414)
(366,420)(367,419)(368,418)(369,417)(370,416)(371,415)(372,421)(373,427)
(374,426)(375,425)(376,424)(377,423)(378,422)(379,428)(380,434)(381,433)
(382,432)(383,431)(384,430)(385,429)(386,435)(387,441)(388,440)(389,439)
(390,438)(391,437)(392,436);
s1 := Sym(441)!( 1, 51)( 2, 50)( 3, 56)( 4, 55)( 5, 54)( 6, 53)( 7, 52)
( 8, 93)( 9, 92)( 10, 98)( 11, 97)( 12, 96)( 13, 95)( 14, 94)( 15, 86)
( 16, 85)( 17, 91)( 18, 90)( 19, 89)( 20, 88)( 21, 87)( 22, 79)( 23, 78)
( 24, 84)( 25, 83)( 26, 82)( 27, 81)( 28, 80)( 29, 72)( 30, 71)( 31, 77)
( 32, 76)( 33, 75)( 34, 74)( 35, 73)( 36, 65)( 37, 64)( 38, 70)( 39, 69)
( 40, 68)( 41, 67)( 42, 66)( 43, 58)( 44, 57)( 45, 63)( 46, 62)( 47, 61)
( 48, 60)( 49, 59)( 99,100)(101,105)(102,104)(106,142)(107,141)(108,147)
(109,146)(110,145)(111,144)(112,143)(113,135)(114,134)(115,140)(116,139)
(117,138)(118,137)(119,136)(120,128)(121,127)(122,133)(123,132)(124,131)
(125,130)(126,129)(148,345)(149,344)(150,350)(151,349)(152,348)(153,347)
(154,346)(155,387)(156,386)(157,392)(158,391)(159,390)(160,389)(161,388)
(162,380)(163,379)(164,385)(165,384)(166,383)(167,382)(168,381)(169,373)
(170,372)(171,378)(172,377)(173,376)(174,375)(175,374)(176,366)(177,365)
(178,371)(179,370)(180,369)(181,368)(182,367)(183,359)(184,358)(185,364)
(186,363)(187,362)(188,361)(189,360)(190,352)(191,351)(192,357)(193,356)
(194,355)(195,354)(196,353)(197,296)(198,295)(199,301)(200,300)(201,299)
(202,298)(203,297)(204,338)(205,337)(206,343)(207,342)(208,341)(209,340)
(210,339)(211,331)(212,330)(213,336)(214,335)(215,334)(216,333)(217,332)
(218,324)(219,323)(220,329)(221,328)(222,327)(223,326)(224,325)(225,317)
(226,316)(227,322)(228,321)(229,320)(230,319)(231,318)(232,310)(233,309)
(234,315)(235,314)(236,313)(237,312)(238,311)(239,303)(240,302)(241,308)
(242,307)(243,306)(244,305)(245,304)(246,394)(247,393)(248,399)(249,398)
(250,397)(251,396)(252,395)(253,436)(254,435)(255,441)(256,440)(257,439)
(258,438)(259,437)(260,429)(261,428)(262,434)(263,433)(264,432)(265,431)
(266,430)(267,422)(268,421)(269,427)(270,426)(271,425)(272,424)(273,423)
(274,415)(275,414)(276,420)(277,419)(278,418)(279,417)(280,416)(281,408)
(282,407)(283,413)(284,412)(285,411)(286,410)(287,409)(288,401)(289,400)
(290,406)(291,405)(292,404)(293,403)(294,402);
s2 := Sym(441)!( 1,155)( 2,161)( 3,160)( 4,159)( 5,158)( 6,157)( 7,156)
( 8,148)( 9,154)( 10,153)( 11,152)( 12,151)( 13,150)( 14,149)( 15,190)
( 16,196)( 17,195)( 18,194)( 19,193)( 20,192)( 21,191)( 22,183)( 23,189)
( 24,188)( 25,187)( 26,186)( 27,185)( 28,184)( 29,176)( 30,182)( 31,181)
( 32,180)( 33,179)( 34,178)( 35,177)( 36,169)( 37,175)( 38,174)( 39,173)
( 40,172)( 41,171)( 42,170)( 43,162)( 44,168)( 45,167)( 46,166)( 47,165)
( 48,164)( 49,163)( 50,253)( 51,259)( 52,258)( 53,257)( 54,256)( 55,255)
( 56,254)( 57,246)( 58,252)( 59,251)( 60,250)( 61,249)( 62,248)( 63,247)
( 64,288)( 65,294)( 66,293)( 67,292)( 68,291)( 69,290)( 70,289)( 71,281)
( 72,287)( 73,286)( 74,285)( 75,284)( 76,283)( 77,282)( 78,274)( 79,280)
( 80,279)( 81,278)( 82,277)( 83,276)( 84,275)( 85,267)( 86,273)( 87,272)
( 88,271)( 89,270)( 90,269)( 91,268)( 92,260)( 93,266)( 94,265)( 95,264)
( 96,263)( 97,262)( 98,261)( 99,204)(100,210)(101,209)(102,208)(103,207)
(104,206)(105,205)(106,197)(107,203)(108,202)(109,201)(110,200)(111,199)
(112,198)(113,239)(114,245)(115,244)(116,243)(117,242)(118,241)(119,240)
(120,232)(121,238)(122,237)(123,236)(124,235)(125,234)(126,233)(127,225)
(128,231)(129,230)(130,229)(131,228)(132,227)(133,226)(134,218)(135,224)
(136,223)(137,222)(138,221)(139,220)(140,219)(141,211)(142,217)(143,216)
(144,215)(145,214)(146,213)(147,212)(295,302)(296,308)(297,307)(298,306)
(299,305)(300,304)(301,303)(309,337)(310,343)(311,342)(312,341)(313,340)
(314,339)(315,338)(316,330)(317,336)(318,335)(319,334)(320,333)(321,332)
(322,331)(324,329)(325,328)(326,327)(344,400)(345,406)(346,405)(347,404)
(348,403)(349,402)(350,401)(351,393)(352,399)(353,398)(354,397)(355,396)
(356,395)(357,394)(358,435)(359,441)(360,440)(361,439)(362,438)(363,437)
(364,436)(365,428)(366,434)(367,433)(368,432)(369,431)(370,430)(371,429)
(372,421)(373,427)(374,426)(375,425)(376,424)(377,423)(378,422)(379,414)
(380,420)(381,419)(382,418)(383,417)(384,416)(385,415)(386,407)(387,413)
(388,412)(389,411)(390,410)(391,409)(392,408);
poly := sub<Sym(441)|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, s2*s0*s1*s0*s1*s2*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 >;
References : None.
to this polytope