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