Polytope of Type {54,6}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,6}*1944a
if this polytope has a name.
Group : SmallGroup(1944,948)
Rank : 3
Schlafli Type : {54,6}
Number of vertices, edges, etc : 162, 486, 18
Order of s0s1s2 : 54
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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) :
2-fold quotients : {27,6}*972a
3-fold quotients : {18,6}*648a, {54,6}*648b
6-fold quotients : {9,6}*324a, {27,6}*324
9-fold quotients : {54,2}*216, {18,6}*216b, {6,6}*216c
18-fold quotients : {27,2}*108, {9,6}*108, {3,6}*108
27-fold quotients : {18,2}*72, {6,6}*72c
54-fold quotients : {9,2}*36, {3,6}*36
81-fold quotients : {6,2}*24
162-fold quotients : {3,2}*12
243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)( 14, 23)
( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 28, 55)( 29, 57)( 30, 56)( 31, 58)
( 32, 60)( 33, 59)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)( 39, 73)
( 40, 78)( 41, 77)( 42, 76)( 43, 81)( 44, 80)( 45, 79)( 46, 66)( 47, 65)
( 48, 64)( 49, 69)( 50, 68)( 51, 67)( 52, 72)( 53, 71)( 54, 70)( 82,183)
( 83,182)( 84,181)( 85,186)( 86,185)( 87,184)( 88,189)( 89,188)( 90,187)
( 91,174)( 92,173)( 93,172)( 94,177)( 95,176)( 96,175)( 97,180)( 98,179)
( 99,178)(100,165)(101,164)(102,163)(103,168)(104,167)(105,166)(106,171)
(107,170)(108,169)(109,237)(110,236)(111,235)(112,240)(113,239)(114,238)
(115,243)(116,242)(117,241)(118,228)(119,227)(120,226)(121,231)(122,230)
(123,229)(124,234)(125,233)(126,232)(127,219)(128,218)(129,217)(130,222)
(131,221)(132,220)(133,225)(134,224)(135,223)(136,210)(137,209)(138,208)
(139,213)(140,212)(141,211)(142,216)(143,215)(144,214)(145,201)(146,200)
(147,199)(148,204)(149,203)(150,202)(151,207)(152,206)(153,205)(154,192)
(155,191)(156,190)(157,195)(158,194)(159,193)(160,198)(161,197)(162,196)
(245,246)(248,249)(251,252)(253,264)(254,263)(255,262)(256,267)(257,266)
(258,265)(259,270)(260,269)(261,268)(271,298)(272,300)(273,299)(274,301)
(275,303)(276,302)(277,304)(278,306)(279,305)(280,318)(281,317)(282,316)
(283,321)(284,320)(285,319)(286,324)(287,323)(288,322)(289,309)(290,308)
(291,307)(292,312)(293,311)(294,310)(295,315)(296,314)(297,313)(325,426)
(326,425)(327,424)(328,429)(329,428)(330,427)(331,432)(332,431)(333,430)
(334,417)(335,416)(336,415)(337,420)(338,419)(339,418)(340,423)(341,422)
(342,421)(343,408)(344,407)(345,406)(346,411)(347,410)(348,409)(349,414)
(350,413)(351,412)(352,480)(353,479)(354,478)(355,483)(356,482)(357,481)
(358,486)(359,485)(360,484)(361,471)(362,470)(363,469)(364,474)(365,473)
(366,472)(367,477)(368,476)(369,475)(370,462)(371,461)(372,460)(373,465)
(374,464)(375,463)(376,468)(377,467)(378,466)(379,453)(380,452)(381,451)
(382,456)(383,455)(384,454)(385,459)(386,458)(387,457)(388,444)(389,443)
(390,442)(391,447)(392,446)(393,445)(394,450)(395,449)(396,448)(397,435)
(398,434)(399,433)(400,438)(401,437)(402,436)(403,441)(404,440)(405,439);;
s1 := ( 1,436)( 2,438)( 3,437)( 4,439)( 5,441)( 6,440)( 7,433)( 8,435)
( 9,434)( 10,456)( 11,455)( 12,454)( 13,459)( 14,458)( 15,457)( 16,453)
( 17,452)( 18,451)( 19,447)( 20,446)( 21,445)( 22,450)( 23,449)( 24,448)
( 25,444)( 26,443)( 27,442)( 28,412)( 29,414)( 30,413)( 31,406)( 32,408)
( 33,407)( 34,409)( 35,411)( 36,410)( 37,432)( 38,431)( 39,430)( 40,426)
( 41,425)( 42,424)( 43,429)( 44,428)( 45,427)( 46,423)( 47,422)( 48,421)
( 49,417)( 50,416)( 51,415)( 52,420)( 53,419)( 54,418)( 55,460)( 56,462)
( 57,461)( 58,463)( 59,465)( 60,464)( 61,466)( 62,468)( 63,467)( 64,480)
( 65,479)( 66,478)( 67,483)( 68,482)( 69,481)( 70,486)( 71,485)( 72,484)
( 73,471)( 74,470)( 75,469)( 76,474)( 77,473)( 78,472)( 79,477)( 80,476)
( 81,475)( 82,355)( 83,357)( 84,356)( 85,358)( 86,360)( 87,359)( 88,352)
( 89,354)( 90,353)( 91,375)( 92,374)( 93,373)( 94,378)( 95,377)( 96,376)
( 97,372)( 98,371)( 99,370)(100,366)(101,365)(102,364)(103,369)(104,368)
(105,367)(106,363)(107,362)(108,361)(109,331)(110,333)(111,332)(112,325)
(113,327)(114,326)(115,328)(116,330)(117,329)(118,351)(119,350)(120,349)
(121,345)(122,344)(123,343)(124,348)(125,347)(126,346)(127,342)(128,341)
(129,340)(130,336)(131,335)(132,334)(133,339)(134,338)(135,337)(136,379)
(137,381)(138,380)(139,382)(140,384)(141,383)(142,385)(143,387)(144,386)
(145,399)(146,398)(147,397)(148,402)(149,401)(150,400)(151,405)(152,404)
(153,403)(154,390)(155,389)(156,388)(157,393)(158,392)(159,391)(160,396)
(161,395)(162,394)(163,274)(164,276)(165,275)(166,277)(167,279)(168,278)
(169,271)(170,273)(171,272)(172,294)(173,293)(174,292)(175,297)(176,296)
(177,295)(178,291)(179,290)(180,289)(181,285)(182,284)(183,283)(184,288)
(185,287)(186,286)(187,282)(188,281)(189,280)(190,250)(191,252)(192,251)
(193,244)(194,246)(195,245)(196,247)(197,249)(198,248)(199,270)(200,269)
(201,268)(202,264)(203,263)(204,262)(205,267)(206,266)(207,265)(208,261)
(209,260)(210,259)(211,255)(212,254)(213,253)(214,258)(215,257)(216,256)
(217,298)(218,300)(219,299)(220,301)(221,303)(222,302)(223,304)(224,306)
(225,305)(226,318)(227,317)(228,316)(229,321)(230,320)(231,319)(232,324)
(233,323)(234,322)(235,309)(236,308)(237,307)(238,312)(239,311)(240,310)
(241,315)(242,314)(243,313);;
s2 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)( 34, 58)
( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)( 42, 72)
( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)( 50, 80)
( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)(185,188)
(186,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)(196,220)
(197,221)(198,222)(199,226)(200,227)(201,228)(202,232)(203,233)(204,234)
(205,229)(206,230)(207,231)(208,235)(209,236)(210,237)(211,241)(212,242)
(213,243)(214,238)(215,239)(216,240)(247,250)(248,251)(249,252)(256,259)
(257,260)(258,261)(265,268)(266,269)(267,270)(271,298)(272,299)(273,300)
(274,304)(275,305)(276,306)(277,301)(278,302)(279,303)(280,307)(281,308)
(282,309)(283,313)(284,314)(285,315)(286,310)(287,311)(288,312)(289,316)
(290,317)(291,318)(292,322)(293,323)(294,324)(295,319)(296,320)(297,321)
(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)(347,350)
(348,351)(352,379)(353,380)(354,381)(355,385)(356,386)(357,387)(358,382)
(359,383)(360,384)(361,388)(362,389)(363,390)(364,394)(365,395)(366,396)
(367,391)(368,392)(369,393)(370,397)(371,398)(372,399)(373,403)(374,404)
(375,405)(376,400)(377,401)(378,402)(409,412)(410,413)(411,414)(418,421)
(419,422)(420,423)(427,430)(428,431)(429,432)(433,460)(434,461)(435,462)
(436,466)(437,467)(438,468)(439,463)(440,464)(441,465)(442,469)(443,470)
(444,471)(445,475)(446,476)(447,477)(448,472)(449,473)(450,474)(451,478)
(452,479)(453,480)(454,484)(455,485)(456,486)(457,481)(458,482)(459,483);;
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*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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*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(486)!( 2, 3)( 5, 6)( 8, 9)( 10, 21)( 11, 20)( 12, 19)( 13, 24)
( 14, 23)( 15, 22)( 16, 27)( 17, 26)( 18, 25)( 28, 55)( 29, 57)( 30, 56)
( 31, 58)( 32, 60)( 33, 59)( 34, 61)( 35, 63)( 36, 62)( 37, 75)( 38, 74)
( 39, 73)( 40, 78)( 41, 77)( 42, 76)( 43, 81)( 44, 80)( 45, 79)( 46, 66)
( 47, 65)( 48, 64)( 49, 69)( 50, 68)( 51, 67)( 52, 72)( 53, 71)( 54, 70)
( 82,183)( 83,182)( 84,181)( 85,186)( 86,185)( 87,184)( 88,189)( 89,188)
( 90,187)( 91,174)( 92,173)( 93,172)( 94,177)( 95,176)( 96,175)( 97,180)
( 98,179)( 99,178)(100,165)(101,164)(102,163)(103,168)(104,167)(105,166)
(106,171)(107,170)(108,169)(109,237)(110,236)(111,235)(112,240)(113,239)
(114,238)(115,243)(116,242)(117,241)(118,228)(119,227)(120,226)(121,231)
(122,230)(123,229)(124,234)(125,233)(126,232)(127,219)(128,218)(129,217)
(130,222)(131,221)(132,220)(133,225)(134,224)(135,223)(136,210)(137,209)
(138,208)(139,213)(140,212)(141,211)(142,216)(143,215)(144,214)(145,201)
(146,200)(147,199)(148,204)(149,203)(150,202)(151,207)(152,206)(153,205)
(154,192)(155,191)(156,190)(157,195)(158,194)(159,193)(160,198)(161,197)
(162,196)(245,246)(248,249)(251,252)(253,264)(254,263)(255,262)(256,267)
(257,266)(258,265)(259,270)(260,269)(261,268)(271,298)(272,300)(273,299)
(274,301)(275,303)(276,302)(277,304)(278,306)(279,305)(280,318)(281,317)
(282,316)(283,321)(284,320)(285,319)(286,324)(287,323)(288,322)(289,309)
(290,308)(291,307)(292,312)(293,311)(294,310)(295,315)(296,314)(297,313)
(325,426)(326,425)(327,424)(328,429)(329,428)(330,427)(331,432)(332,431)
(333,430)(334,417)(335,416)(336,415)(337,420)(338,419)(339,418)(340,423)
(341,422)(342,421)(343,408)(344,407)(345,406)(346,411)(347,410)(348,409)
(349,414)(350,413)(351,412)(352,480)(353,479)(354,478)(355,483)(356,482)
(357,481)(358,486)(359,485)(360,484)(361,471)(362,470)(363,469)(364,474)
(365,473)(366,472)(367,477)(368,476)(369,475)(370,462)(371,461)(372,460)
(373,465)(374,464)(375,463)(376,468)(377,467)(378,466)(379,453)(380,452)
(381,451)(382,456)(383,455)(384,454)(385,459)(386,458)(387,457)(388,444)
(389,443)(390,442)(391,447)(392,446)(393,445)(394,450)(395,449)(396,448)
(397,435)(398,434)(399,433)(400,438)(401,437)(402,436)(403,441)(404,440)
(405,439);
s1 := Sym(486)!( 1,436)( 2,438)( 3,437)( 4,439)( 5,441)( 6,440)( 7,433)
( 8,435)( 9,434)( 10,456)( 11,455)( 12,454)( 13,459)( 14,458)( 15,457)
( 16,453)( 17,452)( 18,451)( 19,447)( 20,446)( 21,445)( 22,450)( 23,449)
( 24,448)( 25,444)( 26,443)( 27,442)( 28,412)( 29,414)( 30,413)( 31,406)
( 32,408)( 33,407)( 34,409)( 35,411)( 36,410)( 37,432)( 38,431)( 39,430)
( 40,426)( 41,425)( 42,424)( 43,429)( 44,428)( 45,427)( 46,423)( 47,422)
( 48,421)( 49,417)( 50,416)( 51,415)( 52,420)( 53,419)( 54,418)( 55,460)
( 56,462)( 57,461)( 58,463)( 59,465)( 60,464)( 61,466)( 62,468)( 63,467)
( 64,480)( 65,479)( 66,478)( 67,483)( 68,482)( 69,481)( 70,486)( 71,485)
( 72,484)( 73,471)( 74,470)( 75,469)( 76,474)( 77,473)( 78,472)( 79,477)
( 80,476)( 81,475)( 82,355)( 83,357)( 84,356)( 85,358)( 86,360)( 87,359)
( 88,352)( 89,354)( 90,353)( 91,375)( 92,374)( 93,373)( 94,378)( 95,377)
( 96,376)( 97,372)( 98,371)( 99,370)(100,366)(101,365)(102,364)(103,369)
(104,368)(105,367)(106,363)(107,362)(108,361)(109,331)(110,333)(111,332)
(112,325)(113,327)(114,326)(115,328)(116,330)(117,329)(118,351)(119,350)
(120,349)(121,345)(122,344)(123,343)(124,348)(125,347)(126,346)(127,342)
(128,341)(129,340)(130,336)(131,335)(132,334)(133,339)(134,338)(135,337)
(136,379)(137,381)(138,380)(139,382)(140,384)(141,383)(142,385)(143,387)
(144,386)(145,399)(146,398)(147,397)(148,402)(149,401)(150,400)(151,405)
(152,404)(153,403)(154,390)(155,389)(156,388)(157,393)(158,392)(159,391)
(160,396)(161,395)(162,394)(163,274)(164,276)(165,275)(166,277)(167,279)
(168,278)(169,271)(170,273)(171,272)(172,294)(173,293)(174,292)(175,297)
(176,296)(177,295)(178,291)(179,290)(180,289)(181,285)(182,284)(183,283)
(184,288)(185,287)(186,286)(187,282)(188,281)(189,280)(190,250)(191,252)
(192,251)(193,244)(194,246)(195,245)(196,247)(197,249)(198,248)(199,270)
(200,269)(201,268)(202,264)(203,263)(204,262)(205,267)(206,266)(207,265)
(208,261)(209,260)(210,259)(211,255)(212,254)(213,253)(214,258)(215,257)
(216,256)(217,298)(218,300)(219,299)(220,301)(221,303)(222,302)(223,304)
(224,306)(225,305)(226,318)(227,317)(228,316)(229,321)(230,320)(231,319)
(232,324)(233,323)(234,322)(235,309)(236,308)(237,307)(238,312)(239,311)
(240,310)(241,315)(242,314)(243,313);
s2 := Sym(486)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 28, 55)( 29, 56)( 30, 57)( 31, 61)( 32, 62)( 33, 63)
( 34, 58)( 35, 59)( 36, 60)( 37, 64)( 38, 65)( 39, 66)( 40, 70)( 41, 71)
( 42, 72)( 43, 67)( 44, 68)( 45, 69)( 46, 73)( 47, 74)( 48, 75)( 49, 79)
( 50, 80)( 51, 81)( 52, 76)( 53, 77)( 54, 78)( 85, 88)( 86, 89)( 87, 90)
( 94, 97)( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(109,136)(110,137)
(111,138)(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)
(119,146)(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)
(127,154)(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)
(135,159)(166,169)(167,170)(168,171)(175,178)(176,179)(177,180)(184,187)
(185,188)(186,189)(190,217)(191,218)(192,219)(193,223)(194,224)(195,225)
(196,220)(197,221)(198,222)(199,226)(200,227)(201,228)(202,232)(203,233)
(204,234)(205,229)(206,230)(207,231)(208,235)(209,236)(210,237)(211,241)
(212,242)(213,243)(214,238)(215,239)(216,240)(247,250)(248,251)(249,252)
(256,259)(257,260)(258,261)(265,268)(266,269)(267,270)(271,298)(272,299)
(273,300)(274,304)(275,305)(276,306)(277,301)(278,302)(279,303)(280,307)
(281,308)(282,309)(283,313)(284,314)(285,315)(286,310)(287,311)(288,312)
(289,316)(290,317)(291,318)(292,322)(293,323)(294,324)(295,319)(296,320)
(297,321)(328,331)(329,332)(330,333)(337,340)(338,341)(339,342)(346,349)
(347,350)(348,351)(352,379)(353,380)(354,381)(355,385)(356,386)(357,387)
(358,382)(359,383)(360,384)(361,388)(362,389)(363,390)(364,394)(365,395)
(366,396)(367,391)(368,392)(369,393)(370,397)(371,398)(372,399)(373,403)
(374,404)(375,405)(376,400)(377,401)(378,402)(409,412)(410,413)(411,414)
(418,421)(419,422)(420,423)(427,430)(428,431)(429,432)(433,460)(434,461)
(435,462)(436,466)(437,467)(438,468)(439,463)(440,464)(441,465)(442,469)
(443,470)(444,471)(445,475)(446,476)(447,477)(448,472)(449,473)(450,474)
(451,478)(452,479)(453,480)(454,484)(455,485)(456,486)(457,481)(458,482)
(459,483);
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, s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s0*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*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