Polytope of Type {6,162}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,162}*1944b
if this polytope has a name.
Group : SmallGroup(1944,955)
Rank : 3
Schlafli Type : {6,162}
Number of vertices, edges, etc : 6, 486, 162
Order of s0s1s2 : 162
Order of s0s1s2s1 : 6
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 : {6,81}*972
3-fold quotients : {2,162}*648, {6,54}*648b
6-fold quotients : {2,81}*324, {6,27}*324
9-fold quotients : {2,54}*216, {6,18}*216b
18-fold quotients : {2,27}*108, {6,9}*108
27-fold quotients : {2,18}*72, {6,6}*72b
54-fold quotients : {2,9}*36, {6,3}*36
81-fold quotients : {2,6}*24
162-fold quotients : {2,3}*12
243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)( 35, 62)
( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)( 43, 70)
( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)( 51, 78)
( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)(113,140)
(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)(121,148)
(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)(129,156)
(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(190,217)(191,218)
(192,219)(193,220)(194,221)(195,222)(196,223)(197,224)(198,225)(199,226)
(200,227)(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)(207,234)
(208,235)(209,236)(210,237)(211,238)(212,239)(213,240)(214,241)(215,242)
(216,243)(271,298)(272,299)(273,300)(274,301)(275,302)(276,303)(277,304)
(278,305)(279,306)(280,307)(281,308)(282,309)(283,310)(284,311)(285,312)
(286,313)(287,314)(288,315)(289,316)(290,317)(291,318)(292,319)(293,320)
(294,321)(295,322)(296,323)(297,324)(352,379)(353,380)(354,381)(355,382)
(356,383)(357,384)(358,385)(359,386)(360,387)(361,388)(362,389)(363,390)
(364,391)(365,392)(366,393)(367,394)(368,395)(369,396)(370,397)(371,398)
(372,399)(373,400)(374,401)(375,402)(376,403)(377,404)(378,405)(433,460)
(434,461)(435,462)(436,463)(437,464)(438,465)(439,466)(440,467)(441,468)
(442,469)(443,470)(444,471)(445,472)(446,473)(447,474)(448,475)(449,476)
(450,477)(451,478)(452,479)(453,480)(454,481)(455,482)(456,483)(457,484)
(458,485)(459,486);;
s1 := ( 1, 28)( 2, 30)( 3, 29)( 4, 35)( 5, 34)( 6, 36)( 7, 32)( 8, 31)
( 9, 33)( 10, 50)( 11, 49)( 12, 51)( 13, 47)( 14, 46)( 15, 48)( 16, 54)
( 17, 53)( 18, 52)( 19, 41)( 20, 40)( 21, 42)( 22, 38)( 23, 37)( 24, 39)
( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 58, 62)( 59, 61)( 60, 63)( 64, 77)
( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)( 72, 79)
( 82,203)( 83,202)( 84,204)( 85,200)( 86,199)( 87,201)( 88,207)( 89,206)
( 90,205)( 91,194)( 92,193)( 93,195)( 94,191)( 95,190)( 96,192)( 97,198)
( 98,197)( 99,196)(100,216)(101,215)(102,214)(103,213)(104,212)(105,211)
(106,210)(107,209)(108,208)(109,176)(110,175)(111,177)(112,173)(113,172)
(114,174)(115,180)(116,179)(117,178)(118,167)(119,166)(120,168)(121,164)
(122,163)(123,165)(124,171)(125,170)(126,169)(127,189)(128,188)(129,187)
(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)(136,230)(137,229)
(138,231)(139,227)(140,226)(141,228)(142,234)(143,233)(144,232)(145,221)
(146,220)(147,222)(148,218)(149,217)(150,219)(151,225)(152,224)(153,223)
(154,243)(155,242)(156,241)(157,240)(158,239)(159,238)(160,237)(161,236)
(162,235)(244,271)(245,273)(246,272)(247,278)(248,277)(249,279)(250,275)
(251,274)(252,276)(253,293)(254,292)(255,294)(256,290)(257,289)(258,291)
(259,297)(260,296)(261,295)(262,284)(263,283)(264,285)(265,281)(266,280)
(267,282)(268,288)(269,287)(270,286)(299,300)(301,305)(302,304)(303,306)
(307,320)(308,319)(309,321)(310,317)(311,316)(312,318)(313,324)(314,323)
(315,322)(325,446)(326,445)(327,447)(328,443)(329,442)(330,444)(331,450)
(332,449)(333,448)(334,437)(335,436)(336,438)(337,434)(338,433)(339,435)
(340,441)(341,440)(342,439)(343,459)(344,458)(345,457)(346,456)(347,455)
(348,454)(349,453)(350,452)(351,451)(352,419)(353,418)(354,420)(355,416)
(356,415)(357,417)(358,423)(359,422)(360,421)(361,410)(362,409)(363,411)
(364,407)(365,406)(366,408)(367,414)(368,413)(369,412)(370,432)(371,431)
(372,430)(373,429)(374,428)(375,427)(376,426)(377,425)(378,424)(379,473)
(380,472)(381,474)(382,470)(383,469)(384,471)(385,477)(386,476)(387,475)
(388,464)(389,463)(390,465)(391,461)(392,460)(393,462)(394,468)(395,467)
(396,466)(397,486)(398,485)(399,484)(400,483)(401,482)(402,481)(403,480)
(404,479)(405,478);;
s2 := ( 1,325)( 2,327)( 3,326)( 4,332)( 5,331)( 6,333)( 7,329)( 8,328)
( 9,330)( 10,347)( 11,346)( 12,348)( 13,344)( 14,343)( 15,345)( 16,351)
( 17,350)( 18,349)( 19,338)( 20,337)( 21,339)( 22,335)( 23,334)( 24,336)
( 25,342)( 26,341)( 27,340)( 28,379)( 29,381)( 30,380)( 31,386)( 32,385)
( 33,387)( 34,383)( 35,382)( 36,384)( 37,401)( 38,400)( 39,402)( 40,398)
( 41,397)( 42,399)( 43,405)( 44,404)( 45,403)( 46,392)( 47,391)( 48,393)
( 49,389)( 50,388)( 51,390)( 52,396)( 53,395)( 54,394)( 55,352)( 56,354)
( 57,353)( 58,359)( 59,358)( 60,360)( 61,356)( 62,355)( 63,357)( 64,374)
( 65,373)( 66,375)( 67,371)( 68,370)( 69,372)( 70,378)( 71,377)( 72,376)
( 73,365)( 74,364)( 75,366)( 76,362)( 77,361)( 78,363)( 79,369)( 80,368)
( 81,367)( 82,244)( 83,246)( 84,245)( 85,251)( 86,250)( 87,252)( 88,248)
( 89,247)( 90,249)( 91,266)( 92,265)( 93,267)( 94,263)( 95,262)( 96,264)
( 97,270)( 98,269)( 99,268)(100,257)(101,256)(102,258)(103,254)(104,253)
(105,255)(106,261)(107,260)(108,259)(109,298)(110,300)(111,299)(112,305)
(113,304)(114,306)(115,302)(116,301)(117,303)(118,320)(119,319)(120,321)
(121,317)(122,316)(123,318)(124,324)(125,323)(126,322)(127,311)(128,310)
(129,312)(130,308)(131,307)(132,309)(133,315)(134,314)(135,313)(136,271)
(137,273)(138,272)(139,278)(140,277)(141,279)(142,275)(143,274)(144,276)
(145,293)(146,292)(147,294)(148,290)(149,289)(150,291)(151,297)(152,296)
(153,295)(154,284)(155,283)(156,285)(157,281)(158,280)(159,282)(160,288)
(161,287)(162,286)(163,419)(164,418)(165,420)(166,416)(167,415)(168,417)
(169,423)(170,422)(171,421)(172,410)(173,409)(174,411)(175,407)(176,406)
(177,408)(178,414)(179,413)(180,412)(181,432)(182,431)(183,430)(184,429)
(185,428)(186,427)(187,426)(188,425)(189,424)(190,473)(191,472)(192,474)
(193,470)(194,469)(195,471)(196,477)(197,476)(198,475)(199,464)(200,463)
(201,465)(202,461)(203,460)(204,462)(205,468)(206,467)(207,466)(208,486)
(209,485)(210,484)(211,483)(212,482)(213,481)(214,480)(215,479)(216,478)
(217,446)(218,445)(219,447)(220,443)(221,442)(222,444)(223,450)(224,449)
(225,448)(226,437)(227,436)(228,438)(229,434)(230,433)(231,435)(232,441)
(233,440)(234,439)(235,459)(236,458)(237,457)(238,456)(239,455)(240,454)
(241,453)(242,452)(243,451);;
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,
s0*s1*s0*s1*s0*s1*s0*s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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(486)!( 28, 55)( 29, 56)( 30, 57)( 31, 58)( 32, 59)( 33, 60)( 34, 61)
( 35, 62)( 36, 63)( 37, 64)( 38, 65)( 39, 66)( 40, 67)( 41, 68)( 42, 69)
( 43, 70)( 44, 71)( 45, 72)( 46, 73)( 47, 74)( 48, 75)( 49, 76)( 50, 77)
( 51, 78)( 52, 79)( 53, 80)( 54, 81)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,145)(119,146)(120,147)
(121,148)(122,149)(123,150)(124,151)(125,152)(126,153)(127,154)(128,155)
(129,156)(130,157)(131,158)(132,159)(133,160)(134,161)(135,162)(190,217)
(191,218)(192,219)(193,220)(194,221)(195,222)(196,223)(197,224)(198,225)
(199,226)(200,227)(201,228)(202,229)(203,230)(204,231)(205,232)(206,233)
(207,234)(208,235)(209,236)(210,237)(211,238)(212,239)(213,240)(214,241)
(215,242)(216,243)(271,298)(272,299)(273,300)(274,301)(275,302)(276,303)
(277,304)(278,305)(279,306)(280,307)(281,308)(282,309)(283,310)(284,311)
(285,312)(286,313)(287,314)(288,315)(289,316)(290,317)(291,318)(292,319)
(293,320)(294,321)(295,322)(296,323)(297,324)(352,379)(353,380)(354,381)
(355,382)(356,383)(357,384)(358,385)(359,386)(360,387)(361,388)(362,389)
(363,390)(364,391)(365,392)(366,393)(367,394)(368,395)(369,396)(370,397)
(371,398)(372,399)(373,400)(374,401)(375,402)(376,403)(377,404)(378,405)
(433,460)(434,461)(435,462)(436,463)(437,464)(438,465)(439,466)(440,467)
(441,468)(442,469)(443,470)(444,471)(445,472)(446,473)(447,474)(448,475)
(449,476)(450,477)(451,478)(452,479)(453,480)(454,481)(455,482)(456,483)
(457,484)(458,485)(459,486);
s1 := Sym(486)!( 1, 28)( 2, 30)( 3, 29)( 4, 35)( 5, 34)( 6, 36)( 7, 32)
( 8, 31)( 9, 33)( 10, 50)( 11, 49)( 12, 51)( 13, 47)( 14, 46)( 15, 48)
( 16, 54)( 17, 53)( 18, 52)( 19, 41)( 20, 40)( 21, 42)( 22, 38)( 23, 37)
( 24, 39)( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 58, 62)( 59, 61)( 60, 63)
( 64, 77)( 65, 76)( 66, 78)( 67, 74)( 68, 73)( 69, 75)( 70, 81)( 71, 80)
( 72, 79)( 82,203)( 83,202)( 84,204)( 85,200)( 86,199)( 87,201)( 88,207)
( 89,206)( 90,205)( 91,194)( 92,193)( 93,195)( 94,191)( 95,190)( 96,192)
( 97,198)( 98,197)( 99,196)(100,216)(101,215)(102,214)(103,213)(104,212)
(105,211)(106,210)(107,209)(108,208)(109,176)(110,175)(111,177)(112,173)
(113,172)(114,174)(115,180)(116,179)(117,178)(118,167)(119,166)(120,168)
(121,164)(122,163)(123,165)(124,171)(125,170)(126,169)(127,189)(128,188)
(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)(136,230)
(137,229)(138,231)(139,227)(140,226)(141,228)(142,234)(143,233)(144,232)
(145,221)(146,220)(147,222)(148,218)(149,217)(150,219)(151,225)(152,224)
(153,223)(154,243)(155,242)(156,241)(157,240)(158,239)(159,238)(160,237)
(161,236)(162,235)(244,271)(245,273)(246,272)(247,278)(248,277)(249,279)
(250,275)(251,274)(252,276)(253,293)(254,292)(255,294)(256,290)(257,289)
(258,291)(259,297)(260,296)(261,295)(262,284)(263,283)(264,285)(265,281)
(266,280)(267,282)(268,288)(269,287)(270,286)(299,300)(301,305)(302,304)
(303,306)(307,320)(308,319)(309,321)(310,317)(311,316)(312,318)(313,324)
(314,323)(315,322)(325,446)(326,445)(327,447)(328,443)(329,442)(330,444)
(331,450)(332,449)(333,448)(334,437)(335,436)(336,438)(337,434)(338,433)
(339,435)(340,441)(341,440)(342,439)(343,459)(344,458)(345,457)(346,456)
(347,455)(348,454)(349,453)(350,452)(351,451)(352,419)(353,418)(354,420)
(355,416)(356,415)(357,417)(358,423)(359,422)(360,421)(361,410)(362,409)
(363,411)(364,407)(365,406)(366,408)(367,414)(368,413)(369,412)(370,432)
(371,431)(372,430)(373,429)(374,428)(375,427)(376,426)(377,425)(378,424)
(379,473)(380,472)(381,474)(382,470)(383,469)(384,471)(385,477)(386,476)
(387,475)(388,464)(389,463)(390,465)(391,461)(392,460)(393,462)(394,468)
(395,467)(396,466)(397,486)(398,485)(399,484)(400,483)(401,482)(402,481)
(403,480)(404,479)(405,478);
s2 := Sym(486)!( 1,325)( 2,327)( 3,326)( 4,332)( 5,331)( 6,333)( 7,329)
( 8,328)( 9,330)( 10,347)( 11,346)( 12,348)( 13,344)( 14,343)( 15,345)
( 16,351)( 17,350)( 18,349)( 19,338)( 20,337)( 21,339)( 22,335)( 23,334)
( 24,336)( 25,342)( 26,341)( 27,340)( 28,379)( 29,381)( 30,380)( 31,386)
( 32,385)( 33,387)( 34,383)( 35,382)( 36,384)( 37,401)( 38,400)( 39,402)
( 40,398)( 41,397)( 42,399)( 43,405)( 44,404)( 45,403)( 46,392)( 47,391)
( 48,393)( 49,389)( 50,388)( 51,390)( 52,396)( 53,395)( 54,394)( 55,352)
( 56,354)( 57,353)( 58,359)( 59,358)( 60,360)( 61,356)( 62,355)( 63,357)
( 64,374)( 65,373)( 66,375)( 67,371)( 68,370)( 69,372)( 70,378)( 71,377)
( 72,376)( 73,365)( 74,364)( 75,366)( 76,362)( 77,361)( 78,363)( 79,369)
( 80,368)( 81,367)( 82,244)( 83,246)( 84,245)( 85,251)( 86,250)( 87,252)
( 88,248)( 89,247)( 90,249)( 91,266)( 92,265)( 93,267)( 94,263)( 95,262)
( 96,264)( 97,270)( 98,269)( 99,268)(100,257)(101,256)(102,258)(103,254)
(104,253)(105,255)(106,261)(107,260)(108,259)(109,298)(110,300)(111,299)
(112,305)(113,304)(114,306)(115,302)(116,301)(117,303)(118,320)(119,319)
(120,321)(121,317)(122,316)(123,318)(124,324)(125,323)(126,322)(127,311)
(128,310)(129,312)(130,308)(131,307)(132,309)(133,315)(134,314)(135,313)
(136,271)(137,273)(138,272)(139,278)(140,277)(141,279)(142,275)(143,274)
(144,276)(145,293)(146,292)(147,294)(148,290)(149,289)(150,291)(151,297)
(152,296)(153,295)(154,284)(155,283)(156,285)(157,281)(158,280)(159,282)
(160,288)(161,287)(162,286)(163,419)(164,418)(165,420)(166,416)(167,415)
(168,417)(169,423)(170,422)(171,421)(172,410)(173,409)(174,411)(175,407)
(176,406)(177,408)(178,414)(179,413)(180,412)(181,432)(182,431)(183,430)
(184,429)(185,428)(186,427)(187,426)(188,425)(189,424)(190,473)(191,472)
(192,474)(193,470)(194,469)(195,471)(196,477)(197,476)(198,475)(199,464)
(200,463)(201,465)(202,461)(203,460)(204,462)(205,468)(206,467)(207,466)
(208,486)(209,485)(210,484)(211,483)(212,482)(213,481)(214,480)(215,479)
(216,478)(217,446)(218,445)(219,447)(220,443)(221,442)(222,444)(223,450)
(224,449)(225,448)(226,437)(227,436)(228,438)(229,434)(230,433)(231,435)
(232,441)(233,440)(234,439)(235,459)(236,458)(237,457)(238,456)(239,455)
(240,454)(241,453)(242,452)(243,451);
poly := sub<Sym(486)|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,
s0*s1*s0*s1*s0*s1*s0*s1*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*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*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