Polytope of Type {54,6}

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