Polytope of Type {9,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {9,6,12}*1296b
if this polytope has a name.
Group : SmallGroup(1296,2020)
Rank : 4
Schlafli Type : {9,6,12}
Number of vertices, edges, etc : 9, 27, 36, 12
Order of s0s1s2s3 : 36
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
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 : {9,6,6}*648b
3-fold quotients : {9,2,12}*432, {9,6,4}*432, {3,6,12}*432b
6-fold quotients : {9,2,6}*216, {9,6,2}*216, {3,6,6}*216b
9-fold quotients : {9,2,4}*144, {3,2,12}*144, {3,6,4}*144
12-fold quotients : {9,2,3}*108
18-fold quotients : {9,2,2}*72, {3,2,6}*72, {3,6,2}*72
27-fold quotients : {3,2,4}*48
36-fold quotients : {3,2,3}*36
54-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,487)( 2,489)( 3,488)( 4,490)( 5,492)( 6,491)( 7,493)( 8,495)
( 9,494)( 10,505)( 11,507)( 12,506)( 13,508)( 14,510)( 15,509)( 16,511)
( 17,513)( 18,512)( 19,496)( 20,498)( 21,497)( 22,499)( 23,501)( 24,500)
( 25,502)( 26,504)( 27,503)( 28,543)( 29,542)( 30,541)( 31,546)( 32,545)
( 33,544)( 34,549)( 35,548)( 36,547)( 37,561)( 38,560)( 39,559)( 40,564)
( 41,563)( 42,562)( 43,567)( 44,566)( 45,565)( 46,552)( 47,551)( 48,550)
( 49,555)( 50,554)( 51,553)( 52,558)( 53,557)( 54,556)( 55,516)( 56,515)
( 57,514)( 58,519)( 59,518)( 60,517)( 61,522)( 62,521)( 63,520)( 64,534)
( 65,533)( 66,532)( 67,537)( 68,536)( 69,535)( 70,540)( 71,539)( 72,538)
( 73,525)( 74,524)( 75,523)( 76,528)( 77,527)( 78,526)( 79,531)( 80,530)
( 81,529)( 82,568)( 83,570)( 84,569)( 85,571)( 86,573)( 87,572)( 88,574)
( 89,576)( 90,575)( 91,586)( 92,588)( 93,587)( 94,589)( 95,591)( 96,590)
( 97,592)( 98,594)( 99,593)(100,577)(101,579)(102,578)(103,580)(104,582)
(105,581)(106,583)(107,585)(108,584)(109,624)(110,623)(111,622)(112,627)
(113,626)(114,625)(115,630)(116,629)(117,628)(118,642)(119,641)(120,640)
(121,645)(122,644)(123,643)(124,648)(125,647)(126,646)(127,633)(128,632)
(129,631)(130,636)(131,635)(132,634)(133,639)(134,638)(135,637)(136,597)
(137,596)(138,595)(139,600)(140,599)(141,598)(142,603)(143,602)(144,601)
(145,615)(146,614)(147,613)(148,618)(149,617)(150,616)(151,621)(152,620)
(153,619)(154,606)(155,605)(156,604)(157,609)(158,608)(159,607)(160,612)
(161,611)(162,610)(163,406)(164,408)(165,407)(166,409)(167,411)(168,410)
(169,412)(170,414)(171,413)(172,424)(173,426)(174,425)(175,427)(176,429)
(177,428)(178,430)(179,432)(180,431)(181,415)(182,417)(183,416)(184,418)
(185,420)(186,419)(187,421)(188,423)(189,422)(190,462)(191,461)(192,460)
(193,465)(194,464)(195,463)(196,468)(197,467)(198,466)(199,480)(200,479)
(201,478)(202,483)(203,482)(204,481)(205,486)(206,485)(207,484)(208,471)
(209,470)(210,469)(211,474)(212,473)(213,472)(214,477)(215,476)(216,475)
(217,435)(218,434)(219,433)(220,438)(221,437)(222,436)(223,441)(224,440)
(225,439)(226,453)(227,452)(228,451)(229,456)(230,455)(231,454)(232,459)
(233,458)(234,457)(235,444)(236,443)(237,442)(238,447)(239,446)(240,445)
(241,450)(242,449)(243,448)(244,325)(245,327)(246,326)(247,328)(248,330)
(249,329)(250,331)(251,333)(252,332)(253,343)(254,345)(255,344)(256,346)
(257,348)(258,347)(259,349)(260,351)(261,350)(262,334)(263,336)(264,335)
(265,337)(266,339)(267,338)(268,340)(269,342)(270,341)(271,381)(272,380)
(273,379)(274,384)(275,383)(276,382)(277,387)(278,386)(279,385)(280,399)
(281,398)(282,397)(283,402)(284,401)(285,400)(286,405)(287,404)(288,403)
(289,390)(290,389)(291,388)(292,393)(293,392)(294,391)(295,396)(296,395)
(297,394)(298,354)(299,353)(300,352)(301,357)(302,356)(303,355)(304,360)
(305,359)(306,358)(307,372)(308,371)(309,370)(310,375)(311,374)(312,373)
(313,378)(314,377)(315,376)(316,363)(317,362)(318,361)(319,366)(320,365)
(321,364)(322,369)(323,368)(324,367);;
s1 := ( 1,523)( 2,525)( 3,524)( 4,526)( 5,528)( 6,527)( 7,529)( 8,531)
( 9,530)( 10,514)( 11,516)( 12,515)( 13,517)( 14,519)( 15,518)( 16,520)
( 17,522)( 18,521)( 19,532)( 20,534)( 21,533)( 22,535)( 23,537)( 24,536)
( 25,538)( 26,540)( 27,539)( 28,496)( 29,498)( 30,497)( 31,499)( 32,501)
( 33,500)( 34,502)( 35,504)( 36,503)( 37,487)( 38,489)( 39,488)( 40,490)
( 41,492)( 42,491)( 43,493)( 44,495)( 45,494)( 46,505)( 47,507)( 48,506)
( 49,508)( 50,510)( 51,509)( 52,511)( 53,513)( 54,512)( 55,552)( 56,551)
( 57,550)( 58,555)( 59,554)( 60,553)( 61,558)( 62,557)( 63,556)( 64,543)
( 65,542)( 66,541)( 67,546)( 68,545)( 69,544)( 70,549)( 71,548)( 72,547)
( 73,561)( 74,560)( 75,559)( 76,564)( 77,563)( 78,562)( 79,567)( 80,566)
( 81,565)( 82,604)( 83,606)( 84,605)( 85,607)( 86,609)( 87,608)( 88,610)
( 89,612)( 90,611)( 91,595)( 92,597)( 93,596)( 94,598)( 95,600)( 96,599)
( 97,601)( 98,603)( 99,602)(100,613)(101,615)(102,614)(103,616)(104,618)
(105,617)(106,619)(107,621)(108,620)(109,577)(110,579)(111,578)(112,580)
(113,582)(114,581)(115,583)(116,585)(117,584)(118,568)(119,570)(120,569)
(121,571)(122,573)(123,572)(124,574)(125,576)(126,575)(127,586)(128,588)
(129,587)(130,589)(131,591)(132,590)(133,592)(134,594)(135,593)(136,633)
(137,632)(138,631)(139,636)(140,635)(141,634)(142,639)(143,638)(144,637)
(145,624)(146,623)(147,622)(148,627)(149,626)(150,625)(151,630)(152,629)
(153,628)(154,642)(155,641)(156,640)(157,645)(158,644)(159,643)(160,648)
(161,647)(162,646)(163,442)(164,444)(165,443)(166,445)(167,447)(168,446)
(169,448)(170,450)(171,449)(172,433)(173,435)(174,434)(175,436)(176,438)
(177,437)(178,439)(179,441)(180,440)(181,451)(182,453)(183,452)(184,454)
(185,456)(186,455)(187,457)(188,459)(189,458)(190,415)(191,417)(192,416)
(193,418)(194,420)(195,419)(196,421)(197,423)(198,422)(199,406)(200,408)
(201,407)(202,409)(203,411)(204,410)(205,412)(206,414)(207,413)(208,424)
(209,426)(210,425)(211,427)(212,429)(213,428)(214,430)(215,432)(216,431)
(217,471)(218,470)(219,469)(220,474)(221,473)(222,472)(223,477)(224,476)
(225,475)(226,462)(227,461)(228,460)(229,465)(230,464)(231,463)(232,468)
(233,467)(234,466)(235,480)(236,479)(237,478)(238,483)(239,482)(240,481)
(241,486)(242,485)(243,484)(244,361)(245,363)(246,362)(247,364)(248,366)
(249,365)(250,367)(251,369)(252,368)(253,352)(254,354)(255,353)(256,355)
(257,357)(258,356)(259,358)(260,360)(261,359)(262,370)(263,372)(264,371)
(265,373)(266,375)(267,374)(268,376)(269,378)(270,377)(271,334)(272,336)
(273,335)(274,337)(275,339)(276,338)(277,340)(278,342)(279,341)(280,325)
(281,327)(282,326)(283,328)(284,330)(285,329)(286,331)(287,333)(288,332)
(289,343)(290,345)(291,344)(292,346)(293,348)(294,347)(295,349)(296,351)
(297,350)(298,390)(299,389)(300,388)(301,393)(302,392)(303,391)(304,396)
(305,395)(306,394)(307,381)(308,380)(309,379)(310,384)(311,383)(312,382)
(313,387)(314,386)(315,385)(316,399)(317,398)(318,397)(319,402)(320,401)
(321,400)(322,405)(323,404)(324,403);;
s2 := ( 1,163)( 2,164)( 3,165)( 4,169)( 5,170)( 6,171)( 7,166)( 8,167)
( 9,168)( 10,181)( 11,182)( 12,183)( 13,187)( 14,188)( 15,189)( 16,184)
( 17,185)( 18,186)( 19,172)( 20,173)( 21,174)( 22,178)( 23,179)( 24,180)
( 25,175)( 26,176)( 27,177)( 28,190)( 29,191)( 30,192)( 31,196)( 32,197)
( 33,198)( 34,193)( 35,194)( 36,195)( 37,208)( 38,209)( 39,210)( 40,214)
( 41,215)( 42,216)( 43,211)( 44,212)( 45,213)( 46,199)( 47,200)( 48,201)
( 49,205)( 50,206)( 51,207)( 52,202)( 53,203)( 54,204)( 55,217)( 56,218)
( 57,219)( 58,223)( 59,224)( 60,225)( 61,220)( 62,221)( 63,222)( 64,235)
( 65,236)( 66,237)( 67,241)( 68,242)( 69,243)( 70,238)( 71,239)( 72,240)
( 73,226)( 74,227)( 75,228)( 76,232)( 77,233)( 78,234)( 79,229)( 80,230)
( 81,231)( 82,244)( 83,245)( 84,246)( 85,250)( 86,251)( 87,252)( 88,247)
( 89,248)( 90,249)( 91,262)( 92,263)( 93,264)( 94,268)( 95,269)( 96,270)
( 97,265)( 98,266)( 99,267)(100,253)(101,254)(102,255)(103,259)(104,260)
(105,261)(106,256)(107,257)(108,258)(109,271)(110,272)(111,273)(112,277)
(113,278)(114,279)(115,274)(116,275)(117,276)(118,289)(119,290)(120,291)
(121,295)(122,296)(123,297)(124,292)(125,293)(126,294)(127,280)(128,281)
(129,282)(130,286)(131,287)(132,288)(133,283)(134,284)(135,285)(136,298)
(137,299)(138,300)(139,304)(140,305)(141,306)(142,301)(143,302)(144,303)
(145,316)(146,317)(147,318)(148,322)(149,323)(150,324)(151,319)(152,320)
(153,321)(154,307)(155,308)(156,309)(157,313)(158,314)(159,315)(160,310)
(161,311)(162,312)(325,568)(326,569)(327,570)(328,574)(329,575)(330,576)
(331,571)(332,572)(333,573)(334,586)(335,587)(336,588)(337,592)(338,593)
(339,594)(340,589)(341,590)(342,591)(343,577)(344,578)(345,579)(346,583)
(347,584)(348,585)(349,580)(350,581)(351,582)(352,595)(353,596)(354,597)
(355,601)(356,602)(357,603)(358,598)(359,599)(360,600)(361,613)(362,614)
(363,615)(364,619)(365,620)(366,621)(367,616)(368,617)(369,618)(370,604)
(371,605)(372,606)(373,610)(374,611)(375,612)(376,607)(377,608)(378,609)
(379,622)(380,623)(381,624)(382,628)(383,629)(384,630)(385,625)(386,626)
(387,627)(388,640)(389,641)(390,642)(391,646)(392,647)(393,648)(394,643)
(395,644)(396,645)(397,631)(398,632)(399,633)(400,637)(401,638)(402,639)
(403,634)(404,635)(405,636)(406,487)(407,488)(408,489)(409,493)(410,494)
(411,495)(412,490)(413,491)(414,492)(415,505)(416,506)(417,507)(418,511)
(419,512)(420,513)(421,508)(422,509)(423,510)(424,496)(425,497)(426,498)
(427,502)(428,503)(429,504)(430,499)(431,500)(432,501)(433,514)(434,515)
(435,516)(436,520)(437,521)(438,522)(439,517)(440,518)(441,519)(442,532)
(443,533)(444,534)(445,538)(446,539)(447,540)(448,535)(449,536)(450,537)
(451,523)(452,524)(453,525)(454,529)(455,530)(456,531)(457,526)(458,527)
(459,528)(460,541)(461,542)(462,543)(463,547)(464,548)(465,549)(466,544)
(467,545)(468,546)(469,559)(470,560)(471,561)(472,565)(473,566)(474,567)
(475,562)(476,563)(477,564)(478,550)(479,551)(480,552)(481,556)(482,557)
(483,558)(484,553)(485,554)(486,555);;
s3 := ( 1, 4)( 2, 5)( 3, 6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)( 20, 23)
( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)( 46, 49)
( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)( 66, 69)
( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)( 92, 95)
( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)(118,121)
(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)(138,141)
(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,247)(164,248)
(165,249)(166,244)(167,245)(168,246)(169,250)(170,251)(171,252)(172,256)
(173,257)(174,258)(175,253)(176,254)(177,255)(178,259)(179,260)(180,261)
(181,265)(182,266)(183,267)(184,262)(185,263)(186,264)(187,268)(188,269)
(189,270)(190,274)(191,275)(192,276)(193,271)(194,272)(195,273)(196,277)
(197,278)(198,279)(199,283)(200,284)(201,285)(202,280)(203,281)(204,282)
(205,286)(206,287)(207,288)(208,292)(209,293)(210,294)(211,289)(212,290)
(213,291)(214,295)(215,296)(216,297)(217,301)(218,302)(219,303)(220,298)
(221,299)(222,300)(223,304)(224,305)(225,306)(226,310)(227,311)(228,312)
(229,307)(230,308)(231,309)(232,313)(233,314)(234,315)(235,319)(236,320)
(237,321)(238,316)(239,317)(240,318)(241,322)(242,323)(243,324)(325,409)
(326,410)(327,411)(328,406)(329,407)(330,408)(331,412)(332,413)(333,414)
(334,418)(335,419)(336,420)(337,415)(338,416)(339,417)(340,421)(341,422)
(342,423)(343,427)(344,428)(345,429)(346,424)(347,425)(348,426)(349,430)
(350,431)(351,432)(352,436)(353,437)(354,438)(355,433)(356,434)(357,435)
(358,439)(359,440)(360,441)(361,445)(362,446)(363,447)(364,442)(365,443)
(366,444)(367,448)(368,449)(369,450)(370,454)(371,455)(372,456)(373,451)
(374,452)(375,453)(376,457)(377,458)(378,459)(379,463)(380,464)(381,465)
(382,460)(383,461)(384,462)(385,466)(386,467)(387,468)(388,472)(389,473)
(390,474)(391,469)(392,470)(393,471)(394,475)(395,476)(396,477)(397,481)
(398,482)(399,483)(400,478)(401,479)(402,480)(403,484)(404,485)(405,486)
(487,490)(488,491)(489,492)(496,499)(497,500)(498,501)(505,508)(506,509)
(507,510)(514,517)(515,518)(516,519)(523,526)(524,527)(525,528)(532,535)
(533,536)(534,537)(541,544)(542,545)(543,546)(550,553)(551,554)(552,555)
(559,562)(560,563)(561,564)(568,571)(569,572)(570,573)(577,580)(578,581)
(579,582)(586,589)(587,590)(588,591)(595,598)(596,599)(597,600)(604,607)
(605,608)(606,609)(613,616)(614,617)(615,618)(622,625)(623,626)(624,627)
(631,634)(632,635)(633,636)(640,643)(641,644)(642,645);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s0*s1*s2*s1*s0*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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(648)!( 1,487)( 2,489)( 3,488)( 4,490)( 5,492)( 6,491)( 7,493)
( 8,495)( 9,494)( 10,505)( 11,507)( 12,506)( 13,508)( 14,510)( 15,509)
( 16,511)( 17,513)( 18,512)( 19,496)( 20,498)( 21,497)( 22,499)( 23,501)
( 24,500)( 25,502)( 26,504)( 27,503)( 28,543)( 29,542)( 30,541)( 31,546)
( 32,545)( 33,544)( 34,549)( 35,548)( 36,547)( 37,561)( 38,560)( 39,559)
( 40,564)( 41,563)( 42,562)( 43,567)( 44,566)( 45,565)( 46,552)( 47,551)
( 48,550)( 49,555)( 50,554)( 51,553)( 52,558)( 53,557)( 54,556)( 55,516)
( 56,515)( 57,514)( 58,519)( 59,518)( 60,517)( 61,522)( 62,521)( 63,520)
( 64,534)( 65,533)( 66,532)( 67,537)( 68,536)( 69,535)( 70,540)( 71,539)
( 72,538)( 73,525)( 74,524)( 75,523)( 76,528)( 77,527)( 78,526)( 79,531)
( 80,530)( 81,529)( 82,568)( 83,570)( 84,569)( 85,571)( 86,573)( 87,572)
( 88,574)( 89,576)( 90,575)( 91,586)( 92,588)( 93,587)( 94,589)( 95,591)
( 96,590)( 97,592)( 98,594)( 99,593)(100,577)(101,579)(102,578)(103,580)
(104,582)(105,581)(106,583)(107,585)(108,584)(109,624)(110,623)(111,622)
(112,627)(113,626)(114,625)(115,630)(116,629)(117,628)(118,642)(119,641)
(120,640)(121,645)(122,644)(123,643)(124,648)(125,647)(126,646)(127,633)
(128,632)(129,631)(130,636)(131,635)(132,634)(133,639)(134,638)(135,637)
(136,597)(137,596)(138,595)(139,600)(140,599)(141,598)(142,603)(143,602)
(144,601)(145,615)(146,614)(147,613)(148,618)(149,617)(150,616)(151,621)
(152,620)(153,619)(154,606)(155,605)(156,604)(157,609)(158,608)(159,607)
(160,612)(161,611)(162,610)(163,406)(164,408)(165,407)(166,409)(167,411)
(168,410)(169,412)(170,414)(171,413)(172,424)(173,426)(174,425)(175,427)
(176,429)(177,428)(178,430)(179,432)(180,431)(181,415)(182,417)(183,416)
(184,418)(185,420)(186,419)(187,421)(188,423)(189,422)(190,462)(191,461)
(192,460)(193,465)(194,464)(195,463)(196,468)(197,467)(198,466)(199,480)
(200,479)(201,478)(202,483)(203,482)(204,481)(205,486)(206,485)(207,484)
(208,471)(209,470)(210,469)(211,474)(212,473)(213,472)(214,477)(215,476)
(216,475)(217,435)(218,434)(219,433)(220,438)(221,437)(222,436)(223,441)
(224,440)(225,439)(226,453)(227,452)(228,451)(229,456)(230,455)(231,454)
(232,459)(233,458)(234,457)(235,444)(236,443)(237,442)(238,447)(239,446)
(240,445)(241,450)(242,449)(243,448)(244,325)(245,327)(246,326)(247,328)
(248,330)(249,329)(250,331)(251,333)(252,332)(253,343)(254,345)(255,344)
(256,346)(257,348)(258,347)(259,349)(260,351)(261,350)(262,334)(263,336)
(264,335)(265,337)(266,339)(267,338)(268,340)(269,342)(270,341)(271,381)
(272,380)(273,379)(274,384)(275,383)(276,382)(277,387)(278,386)(279,385)
(280,399)(281,398)(282,397)(283,402)(284,401)(285,400)(286,405)(287,404)
(288,403)(289,390)(290,389)(291,388)(292,393)(293,392)(294,391)(295,396)
(296,395)(297,394)(298,354)(299,353)(300,352)(301,357)(302,356)(303,355)
(304,360)(305,359)(306,358)(307,372)(308,371)(309,370)(310,375)(311,374)
(312,373)(313,378)(314,377)(315,376)(316,363)(317,362)(318,361)(319,366)
(320,365)(321,364)(322,369)(323,368)(324,367);
s1 := Sym(648)!( 1,523)( 2,525)( 3,524)( 4,526)( 5,528)( 6,527)( 7,529)
( 8,531)( 9,530)( 10,514)( 11,516)( 12,515)( 13,517)( 14,519)( 15,518)
( 16,520)( 17,522)( 18,521)( 19,532)( 20,534)( 21,533)( 22,535)( 23,537)
( 24,536)( 25,538)( 26,540)( 27,539)( 28,496)( 29,498)( 30,497)( 31,499)
( 32,501)( 33,500)( 34,502)( 35,504)( 36,503)( 37,487)( 38,489)( 39,488)
( 40,490)( 41,492)( 42,491)( 43,493)( 44,495)( 45,494)( 46,505)( 47,507)
( 48,506)( 49,508)( 50,510)( 51,509)( 52,511)( 53,513)( 54,512)( 55,552)
( 56,551)( 57,550)( 58,555)( 59,554)( 60,553)( 61,558)( 62,557)( 63,556)
( 64,543)( 65,542)( 66,541)( 67,546)( 68,545)( 69,544)( 70,549)( 71,548)
( 72,547)( 73,561)( 74,560)( 75,559)( 76,564)( 77,563)( 78,562)( 79,567)
( 80,566)( 81,565)( 82,604)( 83,606)( 84,605)( 85,607)( 86,609)( 87,608)
( 88,610)( 89,612)( 90,611)( 91,595)( 92,597)( 93,596)( 94,598)( 95,600)
( 96,599)( 97,601)( 98,603)( 99,602)(100,613)(101,615)(102,614)(103,616)
(104,618)(105,617)(106,619)(107,621)(108,620)(109,577)(110,579)(111,578)
(112,580)(113,582)(114,581)(115,583)(116,585)(117,584)(118,568)(119,570)
(120,569)(121,571)(122,573)(123,572)(124,574)(125,576)(126,575)(127,586)
(128,588)(129,587)(130,589)(131,591)(132,590)(133,592)(134,594)(135,593)
(136,633)(137,632)(138,631)(139,636)(140,635)(141,634)(142,639)(143,638)
(144,637)(145,624)(146,623)(147,622)(148,627)(149,626)(150,625)(151,630)
(152,629)(153,628)(154,642)(155,641)(156,640)(157,645)(158,644)(159,643)
(160,648)(161,647)(162,646)(163,442)(164,444)(165,443)(166,445)(167,447)
(168,446)(169,448)(170,450)(171,449)(172,433)(173,435)(174,434)(175,436)
(176,438)(177,437)(178,439)(179,441)(180,440)(181,451)(182,453)(183,452)
(184,454)(185,456)(186,455)(187,457)(188,459)(189,458)(190,415)(191,417)
(192,416)(193,418)(194,420)(195,419)(196,421)(197,423)(198,422)(199,406)
(200,408)(201,407)(202,409)(203,411)(204,410)(205,412)(206,414)(207,413)
(208,424)(209,426)(210,425)(211,427)(212,429)(213,428)(214,430)(215,432)
(216,431)(217,471)(218,470)(219,469)(220,474)(221,473)(222,472)(223,477)
(224,476)(225,475)(226,462)(227,461)(228,460)(229,465)(230,464)(231,463)
(232,468)(233,467)(234,466)(235,480)(236,479)(237,478)(238,483)(239,482)
(240,481)(241,486)(242,485)(243,484)(244,361)(245,363)(246,362)(247,364)
(248,366)(249,365)(250,367)(251,369)(252,368)(253,352)(254,354)(255,353)
(256,355)(257,357)(258,356)(259,358)(260,360)(261,359)(262,370)(263,372)
(264,371)(265,373)(266,375)(267,374)(268,376)(269,378)(270,377)(271,334)
(272,336)(273,335)(274,337)(275,339)(276,338)(277,340)(278,342)(279,341)
(280,325)(281,327)(282,326)(283,328)(284,330)(285,329)(286,331)(287,333)
(288,332)(289,343)(290,345)(291,344)(292,346)(293,348)(294,347)(295,349)
(296,351)(297,350)(298,390)(299,389)(300,388)(301,393)(302,392)(303,391)
(304,396)(305,395)(306,394)(307,381)(308,380)(309,379)(310,384)(311,383)
(312,382)(313,387)(314,386)(315,385)(316,399)(317,398)(318,397)(319,402)
(320,401)(321,400)(322,405)(323,404)(324,403);
s2 := Sym(648)!( 1,163)( 2,164)( 3,165)( 4,169)( 5,170)( 6,171)( 7,166)
( 8,167)( 9,168)( 10,181)( 11,182)( 12,183)( 13,187)( 14,188)( 15,189)
( 16,184)( 17,185)( 18,186)( 19,172)( 20,173)( 21,174)( 22,178)( 23,179)
( 24,180)( 25,175)( 26,176)( 27,177)( 28,190)( 29,191)( 30,192)( 31,196)
( 32,197)( 33,198)( 34,193)( 35,194)( 36,195)( 37,208)( 38,209)( 39,210)
( 40,214)( 41,215)( 42,216)( 43,211)( 44,212)( 45,213)( 46,199)( 47,200)
( 48,201)( 49,205)( 50,206)( 51,207)( 52,202)( 53,203)( 54,204)( 55,217)
( 56,218)( 57,219)( 58,223)( 59,224)( 60,225)( 61,220)( 62,221)( 63,222)
( 64,235)( 65,236)( 66,237)( 67,241)( 68,242)( 69,243)( 70,238)( 71,239)
( 72,240)( 73,226)( 74,227)( 75,228)( 76,232)( 77,233)( 78,234)( 79,229)
( 80,230)( 81,231)( 82,244)( 83,245)( 84,246)( 85,250)( 86,251)( 87,252)
( 88,247)( 89,248)( 90,249)( 91,262)( 92,263)( 93,264)( 94,268)( 95,269)
( 96,270)( 97,265)( 98,266)( 99,267)(100,253)(101,254)(102,255)(103,259)
(104,260)(105,261)(106,256)(107,257)(108,258)(109,271)(110,272)(111,273)
(112,277)(113,278)(114,279)(115,274)(116,275)(117,276)(118,289)(119,290)
(120,291)(121,295)(122,296)(123,297)(124,292)(125,293)(126,294)(127,280)
(128,281)(129,282)(130,286)(131,287)(132,288)(133,283)(134,284)(135,285)
(136,298)(137,299)(138,300)(139,304)(140,305)(141,306)(142,301)(143,302)
(144,303)(145,316)(146,317)(147,318)(148,322)(149,323)(150,324)(151,319)
(152,320)(153,321)(154,307)(155,308)(156,309)(157,313)(158,314)(159,315)
(160,310)(161,311)(162,312)(325,568)(326,569)(327,570)(328,574)(329,575)
(330,576)(331,571)(332,572)(333,573)(334,586)(335,587)(336,588)(337,592)
(338,593)(339,594)(340,589)(341,590)(342,591)(343,577)(344,578)(345,579)
(346,583)(347,584)(348,585)(349,580)(350,581)(351,582)(352,595)(353,596)
(354,597)(355,601)(356,602)(357,603)(358,598)(359,599)(360,600)(361,613)
(362,614)(363,615)(364,619)(365,620)(366,621)(367,616)(368,617)(369,618)
(370,604)(371,605)(372,606)(373,610)(374,611)(375,612)(376,607)(377,608)
(378,609)(379,622)(380,623)(381,624)(382,628)(383,629)(384,630)(385,625)
(386,626)(387,627)(388,640)(389,641)(390,642)(391,646)(392,647)(393,648)
(394,643)(395,644)(396,645)(397,631)(398,632)(399,633)(400,637)(401,638)
(402,639)(403,634)(404,635)(405,636)(406,487)(407,488)(408,489)(409,493)
(410,494)(411,495)(412,490)(413,491)(414,492)(415,505)(416,506)(417,507)
(418,511)(419,512)(420,513)(421,508)(422,509)(423,510)(424,496)(425,497)
(426,498)(427,502)(428,503)(429,504)(430,499)(431,500)(432,501)(433,514)
(434,515)(435,516)(436,520)(437,521)(438,522)(439,517)(440,518)(441,519)
(442,532)(443,533)(444,534)(445,538)(446,539)(447,540)(448,535)(449,536)
(450,537)(451,523)(452,524)(453,525)(454,529)(455,530)(456,531)(457,526)
(458,527)(459,528)(460,541)(461,542)(462,543)(463,547)(464,548)(465,549)
(466,544)(467,545)(468,546)(469,559)(470,560)(471,561)(472,565)(473,566)
(474,567)(475,562)(476,563)(477,564)(478,550)(479,551)(480,552)(481,556)
(482,557)(483,558)(484,553)(485,554)(486,555);
s3 := Sym(648)!( 1, 4)( 2, 5)( 3, 6)( 10, 13)( 11, 14)( 12, 15)( 19, 22)
( 20, 23)( 21, 24)( 28, 31)( 29, 32)( 30, 33)( 37, 40)( 38, 41)( 39, 42)
( 46, 49)( 47, 50)( 48, 51)( 55, 58)( 56, 59)( 57, 60)( 64, 67)( 65, 68)
( 66, 69)( 73, 76)( 74, 77)( 75, 78)( 82, 85)( 83, 86)( 84, 87)( 91, 94)
( 92, 95)( 93, 96)(100,103)(101,104)(102,105)(109,112)(110,113)(111,114)
(118,121)(119,122)(120,123)(127,130)(128,131)(129,132)(136,139)(137,140)
(138,141)(145,148)(146,149)(147,150)(154,157)(155,158)(156,159)(163,247)
(164,248)(165,249)(166,244)(167,245)(168,246)(169,250)(170,251)(171,252)
(172,256)(173,257)(174,258)(175,253)(176,254)(177,255)(178,259)(179,260)
(180,261)(181,265)(182,266)(183,267)(184,262)(185,263)(186,264)(187,268)
(188,269)(189,270)(190,274)(191,275)(192,276)(193,271)(194,272)(195,273)
(196,277)(197,278)(198,279)(199,283)(200,284)(201,285)(202,280)(203,281)
(204,282)(205,286)(206,287)(207,288)(208,292)(209,293)(210,294)(211,289)
(212,290)(213,291)(214,295)(215,296)(216,297)(217,301)(218,302)(219,303)
(220,298)(221,299)(222,300)(223,304)(224,305)(225,306)(226,310)(227,311)
(228,312)(229,307)(230,308)(231,309)(232,313)(233,314)(234,315)(235,319)
(236,320)(237,321)(238,316)(239,317)(240,318)(241,322)(242,323)(243,324)
(325,409)(326,410)(327,411)(328,406)(329,407)(330,408)(331,412)(332,413)
(333,414)(334,418)(335,419)(336,420)(337,415)(338,416)(339,417)(340,421)
(341,422)(342,423)(343,427)(344,428)(345,429)(346,424)(347,425)(348,426)
(349,430)(350,431)(351,432)(352,436)(353,437)(354,438)(355,433)(356,434)
(357,435)(358,439)(359,440)(360,441)(361,445)(362,446)(363,447)(364,442)
(365,443)(366,444)(367,448)(368,449)(369,450)(370,454)(371,455)(372,456)
(373,451)(374,452)(375,453)(376,457)(377,458)(378,459)(379,463)(380,464)
(381,465)(382,460)(383,461)(384,462)(385,466)(386,467)(387,468)(388,472)
(389,473)(390,474)(391,469)(392,470)(393,471)(394,475)(395,476)(396,477)
(397,481)(398,482)(399,483)(400,478)(401,479)(402,480)(403,484)(404,485)
(405,486)(487,490)(488,491)(489,492)(496,499)(497,500)(498,501)(505,508)
(506,509)(507,510)(514,517)(515,518)(516,519)(523,526)(524,527)(525,528)
(532,535)(533,536)(534,537)(541,544)(542,545)(543,546)(550,553)(551,554)
(552,555)(559,562)(560,563)(561,564)(568,571)(569,572)(570,573)(577,580)
(578,581)(579,582)(586,589)(587,590)(588,591)(595,598)(596,599)(597,600)
(604,607)(605,608)(606,609)(613,616)(614,617)(615,618)(622,625)(623,626)
(624,627)(631,634)(632,635)(633,636)(640,643)(641,644)(642,645);
poly := sub<Sym(648)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s0*s1*s2*s1*s0*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,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope