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# Polytope of Type {6,54}

Atlas Canonical Name : {6,54}*1944e
if this polytope has a name.
Group : SmallGroup(1944,954)
Rank : 3
Schlafli Type : {6,54}
Number of vertices, edges, etc : 18, 486, 162
Order of s0s1s2 : 54
Order of s0s1s2s1 : 6
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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 : {6,27}*972c
3-fold quotients : {6,18}*648a
6-fold quotients : {6,9}*324a
9-fold quotients : {6,18}*216b, {6,6}*216a
18-fold quotients : {6,9}*108, {6,3}*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 := (  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);;
s1 := (  1, 28)(  2, 30)(  3, 29)(  4, 33)(  5, 32)(  6, 31)(  7, 35)(  8, 34)
(  9, 36)( 10, 47)( 11, 46)( 12, 48)( 13, 49)( 14, 51)( 15, 50)( 16, 54)
( 17, 53)( 18, 52)( 19, 38)( 20, 37)( 21, 39)( 22, 40)( 23, 42)( 24, 41)
( 25, 45)( 26, 44)( 27, 43)( 56, 57)( 58, 60)( 61, 62)( 64, 74)( 65, 73)
( 66, 75)( 67, 76)( 68, 78)( 69, 77)( 70, 81)( 71, 80)( 72, 79)( 82,200)
( 83,199)( 84,201)( 85,202)( 86,204)( 87,203)( 88,207)( 89,206)( 90,205)
( 91,191)( 92,190)( 93,192)( 94,193)( 95,195)( 96,194)( 97,198)( 98,197)
( 99,196)(100,210)(101,209)(102,208)(103,212)(104,211)(105,213)(106,214)
(107,216)(108,215)(109,173)(110,172)(111,174)(112,175)(113,177)(114,176)
(115,180)(116,179)(117,178)(118,164)(119,163)(120,165)(121,166)(122,168)
(123,167)(124,171)(125,170)(126,169)(127,183)(128,182)(129,181)(130,185)
(131,184)(132,186)(133,187)(134,189)(135,188)(136,227)(137,226)(138,228)
(139,229)(140,231)(141,230)(142,234)(143,233)(144,232)(145,218)(146,217)
(147,219)(148,220)(149,222)(150,221)(151,225)(152,224)(153,223)(154,237)
(155,236)(156,235)(157,239)(158,238)(159,240)(160,241)(161,243)(162,242)
(244,271)(245,273)(246,272)(247,276)(248,275)(249,274)(250,278)(251,277)
(252,279)(253,290)(254,289)(255,291)(256,292)(257,294)(258,293)(259,297)
(260,296)(261,295)(262,281)(263,280)(264,282)(265,283)(266,285)(267,284)
(268,288)(269,287)(270,286)(299,300)(301,303)(304,305)(307,317)(308,316)
(309,318)(310,319)(311,321)(312,320)(313,324)(314,323)(315,322)(325,443)
(326,442)(327,444)(328,445)(329,447)(330,446)(331,450)(332,449)(333,448)
(334,434)(335,433)(336,435)(337,436)(338,438)(339,437)(340,441)(341,440)
(342,439)(343,453)(344,452)(345,451)(346,455)(347,454)(348,456)(349,457)
(350,459)(351,458)(352,416)(353,415)(354,417)(355,418)(356,420)(357,419)
(358,423)(359,422)(360,421)(361,407)(362,406)(363,408)(364,409)(365,411)
(366,410)(367,414)(368,413)(369,412)(370,426)(371,425)(372,424)(373,428)
(374,427)(375,429)(376,430)(377,432)(378,431)(379,470)(380,469)(381,471)
(382,472)(383,474)(384,473)(385,477)(386,476)(387,475)(388,461)(389,460)
(390,462)(391,463)(392,465)(393,464)(394,468)(395,467)(396,466)(397,480)
(398,479)(399,478)(400,482)(401,481)(402,483)(403,484)(404,486)(405,485);;
s2 := (  1,325)(  2,327)(  3,326)(  4,328)(  5,330)(  6,329)(  7,331)(  8,333)
(  9,332)( 10,344)( 11,343)( 12,345)( 13,347)( 14,346)( 15,348)( 16,350)
( 17,349)( 18,351)( 19,335)( 20,334)( 21,336)( 22,338)( 23,337)( 24,339)
( 25,341)( 26,340)( 27,342)( 28,386)( 29,385)( 30,387)( 31,380)( 32,379)
( 33,381)( 34,383)( 35,382)( 36,384)( 37,405)( 38,404)( 39,403)( 40,399)
( 41,398)( 42,397)( 43,402)( 44,401)( 45,400)( 46,396)( 47,395)( 48,394)
( 49,390)( 50,389)( 51,388)( 52,393)( 53,392)( 54,391)( 55,356)( 56,355)
( 57,357)( 58,359)( 59,358)( 60,360)( 61,353)( 62,352)( 63,354)( 64,375)
( 65,374)( 66,373)( 67,378)( 68,377)( 69,376)( 70,372)( 71,371)( 72,370)
( 73,366)( 74,365)( 75,364)( 76,369)( 77,368)( 78,367)( 79,363)( 80,362)
( 81,361)( 82,244)( 83,246)( 84,245)( 85,247)( 86,249)( 87,248)( 88,250)
( 89,252)( 90,251)( 91,263)( 92,262)( 93,264)( 94,266)( 95,265)( 96,267)
( 97,269)( 98,268)( 99,270)(100,254)(101,253)(102,255)(103,257)(104,256)
(105,258)(106,260)(107,259)(108,261)(109,305)(110,304)(111,306)(112,299)
(113,298)(114,300)(115,302)(116,301)(117,303)(118,324)(119,323)(120,322)
(121,318)(122,317)(123,316)(124,321)(125,320)(126,319)(127,315)(128,314)
(129,313)(130,309)(131,308)(132,307)(133,312)(134,311)(135,310)(136,275)
(137,274)(138,276)(139,278)(140,277)(141,279)(142,272)(143,271)(144,273)
(145,294)(146,293)(147,292)(148,297)(149,296)(150,295)(151,291)(152,290)
(153,289)(154,285)(155,284)(156,283)(157,288)(158,287)(159,286)(160,282)
(161,281)(162,280)(163,416)(164,415)(165,417)(166,419)(167,418)(168,420)
(169,422)(170,421)(171,423)(172,407)(173,406)(174,408)(175,410)(176,409)
(177,411)(178,413)(179,412)(180,414)(181,426)(182,425)(183,424)(184,429)
(185,428)(186,427)(187,432)(188,431)(189,430)(190,477)(191,476)(192,475)
(193,471)(194,470)(195,469)(196,474)(197,473)(198,472)(199,468)(200,467)
(201,466)(202,462)(203,461)(204,460)(205,465)(206,464)(207,463)(208,484)
(209,486)(210,485)(211,478)(212,480)(213,479)(214,481)(215,483)(216,482)
(217,447)(218,446)(219,445)(220,450)(221,449)(222,448)(223,444)(224,443)
(225,442)(226,438)(227,437)(228,436)(229,441)(230,440)(231,439)(232,435)
(233,434)(234,433)(235,454)(236,456)(237,455)(238,457)(239,459)(240,458)
(241,451)(242,453)(243,452);;
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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1,
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s0*s1,
s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2 ];;
poly := F / rels;;

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

```
References : None.
to this polytope