Polytope of Type {54,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,6}*1944d
if this polytope has a name.
Group : SmallGroup(1944,953)
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 : {54,6}*972b
   3-fold quotients : {18,6}*648b
   6-fold quotients : {18,6}*324a
   9-fold quotients : {18,6}*216a, {6,6}*216b
   18-fold quotients : {6,6}*108
   27-fold quotients : {18,2}*72, {6,6}*72a
   54-fold quotients : {9,2}*36
   81-fold quotients : {2,6}*24, {6,2}*24
   162-fold quotients : {2,3}*12, {3,2}*12
   243-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) : 
s0 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 20)( 11, 19)( 12, 21)( 13, 26)
( 14, 25)( 15, 27)( 16, 23)( 17, 22)( 18, 24)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 37, 47)( 38, 46)( 39, 48)( 40, 53)( 41, 52)( 42, 54)( 43, 50)
( 44, 49)( 45, 51)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 74)( 65, 73)
( 66, 75)( 67, 80)( 68, 79)( 69, 81)( 70, 77)( 71, 76)( 72, 78)( 82,173)
( 83,172)( 84,174)( 85,179)( 86,178)( 87,180)( 88,176)( 89,175)( 90,177)
( 91,164)( 92,163)( 93,165)( 94,170)( 95,169)( 96,171)( 97,167)( 98,166)
( 99,168)(100,183)(101,182)(102,181)(103,189)(104,188)(105,187)(106,186)
(107,185)(108,184)(109,200)(110,199)(111,201)(112,206)(113,205)(114,207)
(115,203)(116,202)(117,204)(118,191)(119,190)(120,192)(121,197)(122,196)
(123,198)(124,194)(125,193)(126,195)(127,210)(128,209)(129,208)(130,216)
(131,215)(132,214)(133,213)(134,212)(135,211)(136,227)(137,226)(138,228)
(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)(145,218)(146,217)
(147,219)(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)(154,237)
(155,236)(156,235)(157,243)(158,242)(159,241)(160,240)(161,239)(162,238)
(245,246)(247,250)(248,252)(249,251)(253,263)(254,262)(255,264)(256,269)
(257,268)(258,270)(259,266)(260,265)(261,267)(272,273)(274,277)(275,279)
(276,278)(280,290)(281,289)(282,291)(283,296)(284,295)(285,297)(286,293)
(287,292)(288,294)(299,300)(301,304)(302,306)(303,305)(307,317)(308,316)
(309,318)(310,323)(311,322)(312,324)(313,320)(314,319)(315,321)(325,416)
(326,415)(327,417)(328,422)(329,421)(330,423)(331,419)(332,418)(333,420)
(334,407)(335,406)(336,408)(337,413)(338,412)(339,414)(340,410)(341,409)
(342,411)(343,426)(344,425)(345,424)(346,432)(347,431)(348,430)(349,429)
(350,428)(351,427)(352,443)(353,442)(354,444)(355,449)(356,448)(357,450)
(358,446)(359,445)(360,447)(361,434)(362,433)(363,435)(364,440)(365,439)
(366,441)(367,437)(368,436)(369,438)(370,453)(371,452)(372,451)(373,459)
(374,458)(375,457)(376,456)(377,455)(378,454)(379,470)(380,469)(381,471)
(382,476)(383,475)(384,477)(385,473)(386,472)(387,474)(388,461)(389,460)
(390,462)(391,467)(392,466)(393,468)(394,464)(395,463)(396,465)(397,480)
(398,479)(399,478)(400,486)(401,485)(402,484)(403,483)(404,482)(405,481);;
s1 := (  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)(  8, 90)
(  9, 89)( 10,101)( 11,100)( 12,102)( 13,104)( 14,103)( 15,105)( 16,107)
( 17,106)( 18,108)( 19, 92)( 20, 91)( 21, 93)( 22, 95)( 23, 94)( 24, 96)
( 25, 98)( 26, 97)( 27, 99)( 28,144)( 29,143)( 30,142)( 31,138)( 32,137)
( 33,136)( 34,141)( 35,140)( 36,139)( 37,160)( 38,162)( 39,161)( 40,154)
( 41,156)( 42,155)( 43,157)( 44,159)( 45,158)( 46,151)( 47,153)( 48,152)
( 49,145)( 50,147)( 51,146)( 52,148)( 53,150)( 54,149)( 55,114)( 56,113)
( 57,112)( 58,117)( 59,116)( 60,115)( 61,111)( 62,110)( 63,109)( 64,130)
( 65,132)( 66,131)( 67,133)( 68,135)( 69,134)( 70,127)( 71,129)( 72,128)
( 73,121)( 74,123)( 75,122)( 76,124)( 77,126)( 78,125)( 79,118)( 80,120)
( 81,119)(163,173)(164,172)(165,174)(166,176)(167,175)(168,177)(169,179)
(170,178)(171,180)(181,183)(184,186)(187,189)(190,232)(191,234)(192,233)
(193,226)(194,228)(195,227)(196,229)(197,231)(198,230)(199,223)(200,225)
(201,224)(202,217)(203,219)(204,218)(205,220)(206,222)(207,221)(208,242)
(209,241)(210,243)(211,236)(212,235)(213,237)(214,239)(215,238)(216,240)
(244,325)(245,327)(246,326)(247,328)(248,330)(249,329)(250,331)(251,333)
(252,332)(253,344)(254,343)(255,345)(256,347)(257,346)(258,348)(259,350)
(260,349)(261,351)(262,335)(263,334)(264,336)(265,338)(266,337)(267,339)
(268,341)(269,340)(270,342)(271,387)(272,386)(273,385)(274,381)(275,380)
(276,379)(277,384)(278,383)(279,382)(280,403)(281,405)(282,404)(283,397)
(284,399)(285,398)(286,400)(287,402)(288,401)(289,394)(290,396)(291,395)
(292,388)(293,390)(294,389)(295,391)(296,393)(297,392)(298,357)(299,356)
(300,355)(301,360)(302,359)(303,358)(304,354)(305,353)(306,352)(307,373)
(308,375)(309,374)(310,376)(311,378)(312,377)(313,370)(314,372)(315,371)
(316,364)(317,366)(318,365)(319,367)(320,369)(321,368)(322,361)(323,363)
(324,362)(406,416)(407,415)(408,417)(409,419)(410,418)(411,420)(412,422)
(413,421)(414,423)(424,426)(427,429)(430,432)(433,475)(434,477)(435,476)
(436,469)(437,471)(438,470)(439,472)(440,474)(441,473)(442,466)(443,468)
(444,467)(445,460)(446,462)(447,461)(448,463)(449,465)(450,464)(451,485)
(452,484)(453,486)(454,479)(455,478)(456,480)(457,482)(458,481)(459,483);;
s2 := (  1,271)(  2,272)(  3,273)(  4,279)(  5,277)(  6,278)(  7,275)(  8,276)
(  9,274)( 10,280)( 11,281)( 12,282)( 13,288)( 14,286)( 15,287)( 16,284)
( 17,285)( 18,283)( 19,289)( 20,290)( 21,291)( 22,297)( 23,295)( 24,296)
( 25,293)( 26,294)( 27,292)( 28,244)( 29,245)( 30,246)( 31,252)( 32,250)
( 33,251)( 34,248)( 35,249)( 36,247)( 37,253)( 38,254)( 39,255)( 40,261)
( 41,259)( 42,260)( 43,257)( 44,258)( 45,256)( 46,262)( 47,263)( 48,264)
( 49,270)( 50,268)( 51,269)( 52,266)( 53,267)( 54,265)( 55,298)( 56,299)
( 57,300)( 58,306)( 59,304)( 60,305)( 61,302)( 62,303)( 63,301)( 64,307)
( 65,308)( 66,309)( 67,315)( 68,313)( 69,314)( 70,311)( 71,312)( 72,310)
( 73,316)( 74,317)( 75,318)( 76,324)( 77,322)( 78,323)( 79,320)( 80,321)
( 81,319)( 82,352)( 83,353)( 84,354)( 85,360)( 86,358)( 87,359)( 88,356)
( 89,357)( 90,355)( 91,361)( 92,362)( 93,363)( 94,369)( 95,367)( 96,368)
( 97,365)( 98,366)( 99,364)(100,370)(101,371)(102,372)(103,378)(104,376)
(105,377)(106,374)(107,375)(108,373)(109,325)(110,326)(111,327)(112,333)
(113,331)(114,332)(115,329)(116,330)(117,328)(118,334)(119,335)(120,336)
(121,342)(122,340)(123,341)(124,338)(125,339)(126,337)(127,343)(128,344)
(129,345)(130,351)(131,349)(132,350)(133,347)(134,348)(135,346)(136,379)
(137,380)(138,381)(139,387)(140,385)(141,386)(142,383)(143,384)(144,382)
(145,388)(146,389)(147,390)(148,396)(149,394)(150,395)(151,392)(152,393)
(153,391)(154,397)(155,398)(156,399)(157,405)(158,403)(159,404)(160,401)
(161,402)(162,400)(163,433)(164,434)(165,435)(166,441)(167,439)(168,440)
(169,437)(170,438)(171,436)(172,442)(173,443)(174,444)(175,450)(176,448)
(177,449)(178,446)(179,447)(180,445)(181,451)(182,452)(183,453)(184,459)
(185,457)(186,458)(187,455)(188,456)(189,454)(190,406)(191,407)(192,408)
(193,414)(194,412)(195,413)(196,410)(197,411)(198,409)(199,415)(200,416)
(201,417)(202,423)(203,421)(204,422)(205,419)(206,420)(207,418)(208,424)
(209,425)(210,426)(211,432)(212,430)(213,431)(214,428)(215,429)(216,427)
(217,460)(218,461)(219,462)(220,468)(221,466)(222,467)(223,464)(224,465)
(225,463)(226,469)(227,470)(228,471)(229,477)(230,475)(231,476)(232,473)
(233,474)(234,472)(235,478)(236,479)(237,480)(238,486)(239,484)(240,485)
(241,482)(242,483)(243,481);;
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) : 
s0 := Sym(486)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 10, 20)( 11, 19)( 12, 21)
( 13, 26)( 14, 25)( 15, 27)( 16, 23)( 17, 22)( 18, 24)( 29, 30)( 31, 34)
( 32, 36)( 33, 35)( 37, 47)( 38, 46)( 39, 48)( 40, 53)( 41, 52)( 42, 54)
( 43, 50)( 44, 49)( 45, 51)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 64, 74)
( 65, 73)( 66, 75)( 67, 80)( 68, 79)( 69, 81)( 70, 77)( 71, 76)( 72, 78)
( 82,173)( 83,172)( 84,174)( 85,179)( 86,178)( 87,180)( 88,176)( 89,175)
( 90,177)( 91,164)( 92,163)( 93,165)( 94,170)( 95,169)( 96,171)( 97,167)
( 98,166)( 99,168)(100,183)(101,182)(102,181)(103,189)(104,188)(105,187)
(106,186)(107,185)(108,184)(109,200)(110,199)(111,201)(112,206)(113,205)
(114,207)(115,203)(116,202)(117,204)(118,191)(119,190)(120,192)(121,197)
(122,196)(123,198)(124,194)(125,193)(126,195)(127,210)(128,209)(129,208)
(130,216)(131,215)(132,214)(133,213)(134,212)(135,211)(136,227)(137,226)
(138,228)(139,233)(140,232)(141,234)(142,230)(143,229)(144,231)(145,218)
(146,217)(147,219)(148,224)(149,223)(150,225)(151,221)(152,220)(153,222)
(154,237)(155,236)(156,235)(157,243)(158,242)(159,241)(160,240)(161,239)
(162,238)(245,246)(247,250)(248,252)(249,251)(253,263)(254,262)(255,264)
(256,269)(257,268)(258,270)(259,266)(260,265)(261,267)(272,273)(274,277)
(275,279)(276,278)(280,290)(281,289)(282,291)(283,296)(284,295)(285,297)
(286,293)(287,292)(288,294)(299,300)(301,304)(302,306)(303,305)(307,317)
(308,316)(309,318)(310,323)(311,322)(312,324)(313,320)(314,319)(315,321)
(325,416)(326,415)(327,417)(328,422)(329,421)(330,423)(331,419)(332,418)
(333,420)(334,407)(335,406)(336,408)(337,413)(338,412)(339,414)(340,410)
(341,409)(342,411)(343,426)(344,425)(345,424)(346,432)(347,431)(348,430)
(349,429)(350,428)(351,427)(352,443)(353,442)(354,444)(355,449)(356,448)
(357,450)(358,446)(359,445)(360,447)(361,434)(362,433)(363,435)(364,440)
(365,439)(366,441)(367,437)(368,436)(369,438)(370,453)(371,452)(372,451)
(373,459)(374,458)(375,457)(376,456)(377,455)(378,454)(379,470)(380,469)
(381,471)(382,476)(383,475)(384,477)(385,473)(386,472)(387,474)(388,461)
(389,460)(390,462)(391,467)(392,466)(393,468)(394,464)(395,463)(396,465)
(397,480)(398,479)(399,478)(400,486)(401,485)(402,484)(403,483)(404,482)
(405,481);
s1 := Sym(486)!(  1, 82)(  2, 84)(  3, 83)(  4, 85)(  5, 87)(  6, 86)(  7, 88)
(  8, 90)(  9, 89)( 10,101)( 11,100)( 12,102)( 13,104)( 14,103)( 15,105)
( 16,107)( 17,106)( 18,108)( 19, 92)( 20, 91)( 21, 93)( 22, 95)( 23, 94)
( 24, 96)( 25, 98)( 26, 97)( 27, 99)( 28,144)( 29,143)( 30,142)( 31,138)
( 32,137)( 33,136)( 34,141)( 35,140)( 36,139)( 37,160)( 38,162)( 39,161)
( 40,154)( 41,156)( 42,155)( 43,157)( 44,159)( 45,158)( 46,151)( 47,153)
( 48,152)( 49,145)( 50,147)( 51,146)( 52,148)( 53,150)( 54,149)( 55,114)
( 56,113)( 57,112)( 58,117)( 59,116)( 60,115)( 61,111)( 62,110)( 63,109)
( 64,130)( 65,132)( 66,131)( 67,133)( 68,135)( 69,134)( 70,127)( 71,129)
( 72,128)( 73,121)( 74,123)( 75,122)( 76,124)( 77,126)( 78,125)( 79,118)
( 80,120)( 81,119)(163,173)(164,172)(165,174)(166,176)(167,175)(168,177)
(169,179)(170,178)(171,180)(181,183)(184,186)(187,189)(190,232)(191,234)
(192,233)(193,226)(194,228)(195,227)(196,229)(197,231)(198,230)(199,223)
(200,225)(201,224)(202,217)(203,219)(204,218)(205,220)(206,222)(207,221)
(208,242)(209,241)(210,243)(211,236)(212,235)(213,237)(214,239)(215,238)
(216,240)(244,325)(245,327)(246,326)(247,328)(248,330)(249,329)(250,331)
(251,333)(252,332)(253,344)(254,343)(255,345)(256,347)(257,346)(258,348)
(259,350)(260,349)(261,351)(262,335)(263,334)(264,336)(265,338)(266,337)
(267,339)(268,341)(269,340)(270,342)(271,387)(272,386)(273,385)(274,381)
(275,380)(276,379)(277,384)(278,383)(279,382)(280,403)(281,405)(282,404)
(283,397)(284,399)(285,398)(286,400)(287,402)(288,401)(289,394)(290,396)
(291,395)(292,388)(293,390)(294,389)(295,391)(296,393)(297,392)(298,357)
(299,356)(300,355)(301,360)(302,359)(303,358)(304,354)(305,353)(306,352)
(307,373)(308,375)(309,374)(310,376)(311,378)(312,377)(313,370)(314,372)
(315,371)(316,364)(317,366)(318,365)(319,367)(320,369)(321,368)(322,361)
(323,363)(324,362)(406,416)(407,415)(408,417)(409,419)(410,418)(411,420)
(412,422)(413,421)(414,423)(424,426)(427,429)(430,432)(433,475)(434,477)
(435,476)(436,469)(437,471)(438,470)(439,472)(440,474)(441,473)(442,466)
(443,468)(444,467)(445,460)(446,462)(447,461)(448,463)(449,465)(450,464)
(451,485)(452,484)(453,486)(454,479)(455,478)(456,480)(457,482)(458,481)
(459,483);
s2 := Sym(486)!(  1,271)(  2,272)(  3,273)(  4,279)(  5,277)(  6,278)(  7,275)
(  8,276)(  9,274)( 10,280)( 11,281)( 12,282)( 13,288)( 14,286)( 15,287)
( 16,284)( 17,285)( 18,283)( 19,289)( 20,290)( 21,291)( 22,297)( 23,295)
( 24,296)( 25,293)( 26,294)( 27,292)( 28,244)( 29,245)( 30,246)( 31,252)
( 32,250)( 33,251)( 34,248)( 35,249)( 36,247)( 37,253)( 38,254)( 39,255)
( 40,261)( 41,259)( 42,260)( 43,257)( 44,258)( 45,256)( 46,262)( 47,263)
( 48,264)( 49,270)( 50,268)( 51,269)( 52,266)( 53,267)( 54,265)( 55,298)
( 56,299)( 57,300)( 58,306)( 59,304)( 60,305)( 61,302)( 62,303)( 63,301)
( 64,307)( 65,308)( 66,309)( 67,315)( 68,313)( 69,314)( 70,311)( 71,312)
( 72,310)( 73,316)( 74,317)( 75,318)( 76,324)( 77,322)( 78,323)( 79,320)
( 80,321)( 81,319)( 82,352)( 83,353)( 84,354)( 85,360)( 86,358)( 87,359)
( 88,356)( 89,357)( 90,355)( 91,361)( 92,362)( 93,363)( 94,369)( 95,367)
( 96,368)( 97,365)( 98,366)( 99,364)(100,370)(101,371)(102,372)(103,378)
(104,376)(105,377)(106,374)(107,375)(108,373)(109,325)(110,326)(111,327)
(112,333)(113,331)(114,332)(115,329)(116,330)(117,328)(118,334)(119,335)
(120,336)(121,342)(122,340)(123,341)(124,338)(125,339)(126,337)(127,343)
(128,344)(129,345)(130,351)(131,349)(132,350)(133,347)(134,348)(135,346)
(136,379)(137,380)(138,381)(139,387)(140,385)(141,386)(142,383)(143,384)
(144,382)(145,388)(146,389)(147,390)(148,396)(149,394)(150,395)(151,392)
(152,393)(153,391)(154,397)(155,398)(156,399)(157,405)(158,403)(159,404)
(160,401)(161,402)(162,400)(163,433)(164,434)(165,435)(166,441)(167,439)
(168,440)(169,437)(170,438)(171,436)(172,442)(173,443)(174,444)(175,450)
(176,448)(177,449)(178,446)(179,447)(180,445)(181,451)(182,452)(183,453)
(184,459)(185,457)(186,458)(187,455)(188,456)(189,454)(190,406)(191,407)
(192,408)(193,414)(194,412)(195,413)(196,410)(197,411)(198,409)(199,415)
(200,416)(201,417)(202,423)(203,421)(204,422)(205,419)(206,420)(207,418)
(208,424)(209,425)(210,426)(211,432)(212,430)(213,431)(214,428)(215,429)
(216,427)(217,460)(218,461)(219,462)(220,468)(221,466)(222,467)(223,464)
(224,465)(225,463)(226,469)(227,470)(228,471)(229,477)(230,475)(231,476)
(232,473)(233,474)(234,472)(235,478)(236,479)(237,480)(238,486)(239,484)
(240,485)(241,482)(242,483)(243,481);
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, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s2*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s2*s1*s0*s1*s0*s2*s1*s0*s2*s1*s0*s1*s0*s1*s0*s1 >;
References : None.
to this polytope