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Polytope of Type {3,6,36}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,36}*1296b
if this polytope has a name.
Group : SmallGroup(1296,2007)
Rank : 4
Schlafli Type : {3,6,36}
Number of vertices, edges, etc : 3, 9, 108, 36
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 : {3,6,18}*648b
   3-fold quotients : {3,2,36}*432, {3,6,12}*432b
   6-fold quotients : {3,2,18}*216, {3,6,6}*216b
   9-fold quotients : {3,2,12}*144, {3,6,4}*144
   12-fold quotients : {3,2,9}*108
   18-fold quotients : {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)(  2,488)(  3,489)(  4,490)(  5,491)(  6,492)(  7,493)(  8,494)
(  9,495)( 10,505)( 11,506)( 12,507)( 13,508)( 14,509)( 15,510)( 16,511)
( 17,512)( 18,513)( 19,496)( 20,497)( 21,498)( 22,499)( 23,500)( 24,501)
( 25,502)( 26,503)( 27,504)( 28,541)( 29,542)( 30,543)( 31,544)( 32,545)
( 33,546)( 34,547)( 35,548)( 36,549)( 37,559)( 38,560)( 39,561)( 40,562)
( 41,563)( 42,564)( 43,565)( 44,566)( 45,567)( 46,550)( 47,551)( 48,552)
( 49,553)( 50,554)( 51,555)( 52,556)( 53,557)( 54,558)( 55,514)( 56,515)
( 57,516)( 58,517)( 59,518)( 60,519)( 61,520)( 62,521)( 63,522)( 64,532)
( 65,533)( 66,534)( 67,535)( 68,536)( 69,537)( 70,538)( 71,539)( 72,540)
( 73,523)( 74,524)( 75,525)( 76,526)( 77,527)( 78,528)( 79,529)( 80,530)
( 81,531)( 82,568)( 83,569)( 84,570)( 85,571)( 86,572)( 87,573)( 88,574)
( 89,575)( 90,576)( 91,586)( 92,587)( 93,588)( 94,589)( 95,590)( 96,591)
( 97,592)( 98,593)( 99,594)(100,577)(101,578)(102,579)(103,580)(104,581)
(105,582)(106,583)(107,584)(108,585)(109,622)(110,623)(111,624)(112,625)
(113,626)(114,627)(115,628)(116,629)(117,630)(118,640)(119,641)(120,642)
(121,643)(122,644)(123,645)(124,646)(125,647)(126,648)(127,631)(128,632)
(129,633)(130,634)(131,635)(132,636)(133,637)(134,638)(135,639)(136,595)
(137,596)(138,597)(139,598)(140,599)(141,600)(142,601)(143,602)(144,603)
(145,613)(146,614)(147,615)(148,616)(149,617)(150,618)(151,619)(152,620)
(153,621)(154,604)(155,605)(156,606)(157,607)(158,608)(159,609)(160,610)
(161,611)(162,612)(163,406)(164,407)(165,408)(166,409)(167,410)(168,411)
(169,412)(170,413)(171,414)(172,424)(173,425)(174,426)(175,427)(176,428)
(177,429)(178,430)(179,431)(180,432)(181,415)(182,416)(183,417)(184,418)
(185,419)(186,420)(187,421)(188,422)(189,423)(190,460)(191,461)(192,462)
(193,463)(194,464)(195,465)(196,466)(197,467)(198,468)(199,478)(200,479)
(201,480)(202,481)(203,482)(204,483)(205,484)(206,485)(207,486)(208,469)
(209,470)(210,471)(211,472)(212,473)(213,474)(214,475)(215,476)(216,477)
(217,433)(218,434)(219,435)(220,436)(221,437)(222,438)(223,439)(224,440)
(225,441)(226,451)(227,452)(228,453)(229,454)(230,455)(231,456)(232,457)
(233,458)(234,459)(235,442)(236,443)(237,444)(238,445)(239,446)(240,447)
(241,448)(242,449)(243,450)(244,325)(245,326)(246,327)(247,328)(248,329)
(249,330)(250,331)(251,332)(252,333)(253,343)(254,344)(255,345)(256,346)
(257,347)(258,348)(259,349)(260,350)(261,351)(262,334)(263,335)(264,336)
(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,379)(272,380)
(273,381)(274,382)(275,383)(276,384)(277,385)(278,386)(279,387)(280,397)
(281,398)(282,399)(283,400)(284,401)(285,402)(286,403)(287,404)(288,405)
(289,388)(290,389)(291,390)(292,391)(293,392)(294,393)(295,394)(296,395)
(297,396)(298,352)(299,353)(300,354)(301,355)(302,356)(303,357)(304,358)
(305,359)(306,360)(307,370)(308,371)(309,372)(310,373)(311,374)(312,375)
(313,376)(314,377)(315,378)(316,361)(317,362)(318,363)(319,364)(320,365)
(321,366)(322,367)(323,368)(324,369);;
s1 := (  1,523)(  2,524)(  3,525)(  4,526)(  5,527)(  6,528)(  7,529)(  8,530)
(  9,531)( 10,514)( 11,515)( 12,516)( 13,517)( 14,518)( 15,519)( 16,520)
( 17,521)( 18,522)( 19,532)( 20,533)( 21,534)( 22,535)( 23,536)( 24,537)
( 25,538)( 26,539)( 27,540)( 28,496)( 29,497)( 30,498)( 31,499)( 32,500)
( 33,501)( 34,502)( 35,503)( 36,504)( 37,487)( 38,488)( 39,489)( 40,490)
( 41,491)( 42,492)( 43,493)( 44,494)( 45,495)( 46,505)( 47,506)( 48,507)
( 49,508)( 50,509)( 51,510)( 52,511)( 53,512)( 54,513)( 55,550)( 56,551)
( 57,552)( 58,553)( 59,554)( 60,555)( 61,556)( 62,557)( 63,558)( 64,541)
( 65,542)( 66,543)( 67,544)( 68,545)( 69,546)( 70,547)( 71,548)( 72,549)
( 73,559)( 74,560)( 75,561)( 76,562)( 77,563)( 78,564)( 79,565)( 80,566)
( 81,567)( 82,604)( 83,605)( 84,606)( 85,607)( 86,608)( 87,609)( 88,610)
( 89,611)( 90,612)( 91,595)( 92,596)( 93,597)( 94,598)( 95,599)( 96,600)
( 97,601)( 98,602)( 99,603)(100,613)(101,614)(102,615)(103,616)(104,617)
(105,618)(106,619)(107,620)(108,621)(109,577)(110,578)(111,579)(112,580)
(113,581)(114,582)(115,583)(116,584)(117,585)(118,568)(119,569)(120,570)
(121,571)(122,572)(123,573)(124,574)(125,575)(126,576)(127,586)(128,587)
(129,588)(130,589)(131,590)(132,591)(133,592)(134,593)(135,594)(136,631)
(137,632)(138,633)(139,634)(140,635)(141,636)(142,637)(143,638)(144,639)
(145,622)(146,623)(147,624)(148,625)(149,626)(150,627)(151,628)(152,629)
(153,630)(154,640)(155,641)(156,642)(157,643)(158,644)(159,645)(160,646)
(161,647)(162,648)(163,442)(164,443)(165,444)(166,445)(167,446)(168,447)
(169,448)(170,449)(171,450)(172,433)(173,434)(174,435)(175,436)(176,437)
(177,438)(178,439)(179,440)(180,441)(181,451)(182,452)(183,453)(184,454)
(185,455)(186,456)(187,457)(188,458)(189,459)(190,415)(191,416)(192,417)
(193,418)(194,419)(195,420)(196,421)(197,422)(198,423)(199,406)(200,407)
(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414)(208,424)
(209,425)(210,426)(211,427)(212,428)(213,429)(214,430)(215,431)(216,432)
(217,469)(218,470)(219,471)(220,472)(221,473)(222,474)(223,475)(224,476)
(225,477)(226,460)(227,461)(228,462)(229,463)(230,464)(231,465)(232,466)
(233,467)(234,468)(235,478)(236,479)(237,480)(238,481)(239,482)(240,483)
(241,484)(242,485)(243,486)(244,361)(245,362)(246,363)(247,364)(248,365)
(249,366)(250,367)(251,368)(252,369)(253,352)(254,353)(255,354)(256,355)
(257,356)(258,357)(259,358)(260,359)(261,360)(262,370)(263,371)(264,372)
(265,373)(266,374)(267,375)(268,376)(269,377)(270,378)(271,334)(272,335)
(273,336)(274,337)(275,338)(276,339)(277,340)(278,341)(279,342)(280,325)
(281,326)(282,327)(283,328)(284,329)(285,330)(286,331)(287,332)(288,333)
(289,343)(290,344)(291,345)(292,346)(293,347)(294,348)(295,349)(296,350)
(297,351)(298,388)(299,389)(300,390)(301,391)(302,392)(303,393)(304,394)
(305,395)(306,396)(307,379)(308,380)(309,381)(310,382)(311,383)(312,384)
(313,385)(314,386)(315,387)(316,397)(317,398)(318,399)(319,400)(320,401)
(321,402)(322,403)(323,404)(324,405);;
s2 := (  1,325)(  2,327)(  3,326)(  4,333)(  5,332)(  6,331)(  7,330)(  8,329)
(  9,328)( 10,334)( 11,336)( 12,335)( 13,342)( 14,341)( 15,340)( 16,339)
( 17,338)( 18,337)( 19,343)( 20,345)( 21,344)( 22,351)( 23,350)( 24,349)
( 25,348)( 26,347)( 27,346)( 28,379)( 29,381)( 30,380)( 31,387)( 32,386)
( 33,385)( 34,384)( 35,383)( 36,382)( 37,388)( 38,390)( 39,389)( 40,396)
( 41,395)( 42,394)( 43,393)( 44,392)( 45,391)( 46,397)( 47,399)( 48,398)
( 49,405)( 50,404)( 51,403)( 52,402)( 53,401)( 54,400)( 55,352)( 56,354)
( 57,353)( 58,360)( 59,359)( 60,358)( 61,357)( 62,356)( 63,355)( 64,361)
( 65,363)( 66,362)( 67,369)( 68,368)( 69,367)( 70,366)( 71,365)( 72,364)
( 73,370)( 74,372)( 75,371)( 76,378)( 77,377)( 78,376)( 79,375)( 80,374)
( 81,373)( 82,406)( 83,408)( 84,407)( 85,414)( 86,413)( 87,412)( 88,411)
( 89,410)( 90,409)( 91,415)( 92,417)( 93,416)( 94,423)( 95,422)( 96,421)
( 97,420)( 98,419)( 99,418)(100,424)(101,426)(102,425)(103,432)(104,431)
(105,430)(106,429)(107,428)(108,427)(109,460)(110,462)(111,461)(112,468)
(113,467)(114,466)(115,465)(116,464)(117,463)(118,469)(119,471)(120,470)
(121,477)(122,476)(123,475)(124,474)(125,473)(126,472)(127,478)(128,480)
(129,479)(130,486)(131,485)(132,484)(133,483)(134,482)(135,481)(136,433)
(137,435)(138,434)(139,441)(140,440)(141,439)(142,438)(143,437)(144,436)
(145,442)(146,444)(147,443)(148,450)(149,449)(150,448)(151,447)(152,446)
(153,445)(154,451)(155,453)(156,452)(157,459)(158,458)(159,457)(160,456)
(161,455)(162,454)(163,568)(164,570)(165,569)(166,576)(167,575)(168,574)
(169,573)(170,572)(171,571)(172,577)(173,579)(174,578)(175,585)(176,584)
(177,583)(178,582)(179,581)(180,580)(181,586)(182,588)(183,587)(184,594)
(185,593)(186,592)(187,591)(188,590)(189,589)(190,622)(191,624)(192,623)
(193,630)(194,629)(195,628)(196,627)(197,626)(198,625)(199,631)(200,633)
(201,632)(202,639)(203,638)(204,637)(205,636)(206,635)(207,634)(208,640)
(209,642)(210,641)(211,648)(212,647)(213,646)(214,645)(215,644)(216,643)
(217,595)(218,597)(219,596)(220,603)(221,602)(222,601)(223,600)(224,599)
(225,598)(226,604)(227,606)(228,605)(229,612)(230,611)(231,610)(232,609)
(233,608)(234,607)(235,613)(236,615)(237,614)(238,621)(239,620)(240,619)
(241,618)(242,617)(243,616)(244,487)(245,489)(246,488)(247,495)(248,494)
(249,493)(250,492)(251,491)(252,490)(253,496)(254,498)(255,497)(256,504)
(257,503)(258,502)(259,501)(260,500)(261,499)(262,505)(263,507)(264,506)
(265,513)(266,512)(267,511)(268,510)(269,509)(270,508)(271,541)(272,543)
(273,542)(274,549)(275,548)(276,547)(277,546)(278,545)(279,544)(280,550)
(281,552)(282,551)(283,558)(284,557)(285,556)(286,555)(287,554)(288,553)
(289,559)(290,561)(291,560)(292,567)(293,566)(294,565)(295,564)(296,563)
(297,562)(298,514)(299,516)(300,515)(301,522)(302,521)(303,520)(304,519)
(305,518)(306,517)(307,523)(308,525)(309,524)(310,531)(311,530)(312,529)
(313,528)(314,527)(315,526)(316,532)(317,534)(318,533)(319,540)(320,539)
(321,538)(322,537)(323,536)(324,535);;
s3 := (  1,  4)(  2,  6)(  3,  5)(  7,  9)( 10, 13)( 11, 15)( 12, 14)( 16, 18)
( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 31)( 29, 33)( 30, 32)( 34, 36)
( 37, 40)( 38, 42)( 39, 41)( 43, 45)( 46, 49)( 47, 51)( 48, 50)( 52, 54)
( 55, 58)( 56, 60)( 57, 59)( 61, 63)( 64, 67)( 65, 69)( 66, 68)( 70, 72)
( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 85)( 83, 87)( 84, 86)( 88, 90)
( 91, 94)( 92, 96)( 93, 95)( 97, 99)(100,103)(101,105)(102,104)(106,108)
(109,112)(110,114)(111,113)(115,117)(118,121)(119,123)(120,122)(124,126)
(127,130)(128,132)(129,131)(133,135)(136,139)(137,141)(138,140)(142,144)
(145,148)(146,150)(147,149)(151,153)(154,157)(155,159)(156,158)(160,162)
(163,247)(164,249)(165,248)(166,244)(167,246)(168,245)(169,252)(170,251)
(171,250)(172,256)(173,258)(174,257)(175,253)(176,255)(177,254)(178,261)
(179,260)(180,259)(181,265)(182,267)(183,266)(184,262)(185,264)(186,263)
(187,270)(188,269)(189,268)(190,274)(191,276)(192,275)(193,271)(194,273)
(195,272)(196,279)(197,278)(198,277)(199,283)(200,285)(201,284)(202,280)
(203,282)(204,281)(205,288)(206,287)(207,286)(208,292)(209,294)(210,293)
(211,289)(212,291)(213,290)(214,297)(215,296)(216,295)(217,301)(218,303)
(219,302)(220,298)(221,300)(222,299)(223,306)(224,305)(225,304)(226,310)
(227,312)(228,311)(229,307)(230,309)(231,308)(232,315)(233,314)(234,313)
(235,319)(236,321)(237,320)(238,316)(239,318)(240,317)(241,324)(242,323)
(243,322)(325,409)(326,411)(327,410)(328,406)(329,408)(330,407)(331,414)
(332,413)(333,412)(334,418)(335,420)(336,419)(337,415)(338,417)(339,416)
(340,423)(341,422)(342,421)(343,427)(344,429)(345,428)(346,424)(347,426)
(348,425)(349,432)(350,431)(351,430)(352,436)(353,438)(354,437)(355,433)
(356,435)(357,434)(358,441)(359,440)(360,439)(361,445)(362,447)(363,446)
(364,442)(365,444)(366,443)(367,450)(368,449)(369,448)(370,454)(371,456)
(372,455)(373,451)(374,453)(375,452)(376,459)(377,458)(378,457)(379,463)
(380,465)(381,464)(382,460)(383,462)(384,461)(385,468)(386,467)(387,466)
(388,472)(389,474)(390,473)(391,469)(392,471)(393,470)(394,477)(395,476)
(396,475)(397,481)(398,483)(399,482)(400,478)(401,480)(402,479)(403,486)
(404,485)(405,484)(487,490)(488,492)(489,491)(493,495)(496,499)(497,501)
(498,500)(502,504)(505,508)(506,510)(507,509)(511,513)(514,517)(515,519)
(516,518)(520,522)(523,526)(524,528)(525,527)(529,531)(532,535)(533,537)
(534,536)(538,540)(541,544)(542,546)(543,545)(547,549)(550,553)(551,555)
(552,554)(556,558)(559,562)(560,564)(561,563)(565,567)(568,571)(569,573)
(570,572)(574,576)(577,580)(578,582)(579,581)(583,585)(586,589)(587,591)
(588,590)(592,594)(595,598)(596,600)(597,599)(601,603)(604,607)(605,609)
(606,608)(610,612)(613,616)(614,618)(615,617)(619,621)(622,625)(623,627)
(624,626)(628,630)(631,634)(632,636)(633,635)(637,639)(640,643)(641,645)
(642,644)(646,648);;
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, s0*s1*s0*s1*s0*s1, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s2*s1*s2*s0*s1*s2*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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)(  2,488)(  3,489)(  4,490)(  5,491)(  6,492)(  7,493)
(  8,494)(  9,495)( 10,505)( 11,506)( 12,507)( 13,508)( 14,509)( 15,510)
( 16,511)( 17,512)( 18,513)( 19,496)( 20,497)( 21,498)( 22,499)( 23,500)
( 24,501)( 25,502)( 26,503)( 27,504)( 28,541)( 29,542)( 30,543)( 31,544)
( 32,545)( 33,546)( 34,547)( 35,548)( 36,549)( 37,559)( 38,560)( 39,561)
( 40,562)( 41,563)( 42,564)( 43,565)( 44,566)( 45,567)( 46,550)( 47,551)
( 48,552)( 49,553)( 50,554)( 51,555)( 52,556)( 53,557)( 54,558)( 55,514)
( 56,515)( 57,516)( 58,517)( 59,518)( 60,519)( 61,520)( 62,521)( 63,522)
( 64,532)( 65,533)( 66,534)( 67,535)( 68,536)( 69,537)( 70,538)( 71,539)
( 72,540)( 73,523)( 74,524)( 75,525)( 76,526)( 77,527)( 78,528)( 79,529)
( 80,530)( 81,531)( 82,568)( 83,569)( 84,570)( 85,571)( 86,572)( 87,573)
( 88,574)( 89,575)( 90,576)( 91,586)( 92,587)( 93,588)( 94,589)( 95,590)
( 96,591)( 97,592)( 98,593)( 99,594)(100,577)(101,578)(102,579)(103,580)
(104,581)(105,582)(106,583)(107,584)(108,585)(109,622)(110,623)(111,624)
(112,625)(113,626)(114,627)(115,628)(116,629)(117,630)(118,640)(119,641)
(120,642)(121,643)(122,644)(123,645)(124,646)(125,647)(126,648)(127,631)
(128,632)(129,633)(130,634)(131,635)(132,636)(133,637)(134,638)(135,639)
(136,595)(137,596)(138,597)(139,598)(140,599)(141,600)(142,601)(143,602)
(144,603)(145,613)(146,614)(147,615)(148,616)(149,617)(150,618)(151,619)
(152,620)(153,621)(154,604)(155,605)(156,606)(157,607)(158,608)(159,609)
(160,610)(161,611)(162,612)(163,406)(164,407)(165,408)(166,409)(167,410)
(168,411)(169,412)(170,413)(171,414)(172,424)(173,425)(174,426)(175,427)
(176,428)(177,429)(178,430)(179,431)(180,432)(181,415)(182,416)(183,417)
(184,418)(185,419)(186,420)(187,421)(188,422)(189,423)(190,460)(191,461)
(192,462)(193,463)(194,464)(195,465)(196,466)(197,467)(198,468)(199,478)
(200,479)(201,480)(202,481)(203,482)(204,483)(205,484)(206,485)(207,486)
(208,469)(209,470)(210,471)(211,472)(212,473)(213,474)(214,475)(215,476)
(216,477)(217,433)(218,434)(219,435)(220,436)(221,437)(222,438)(223,439)
(224,440)(225,441)(226,451)(227,452)(228,453)(229,454)(230,455)(231,456)
(232,457)(233,458)(234,459)(235,442)(236,443)(237,444)(238,445)(239,446)
(240,447)(241,448)(242,449)(243,450)(244,325)(245,326)(246,327)(247,328)
(248,329)(249,330)(250,331)(251,332)(252,333)(253,343)(254,344)(255,345)
(256,346)(257,347)(258,348)(259,349)(260,350)(261,351)(262,334)(263,335)
(264,336)(265,337)(266,338)(267,339)(268,340)(269,341)(270,342)(271,379)
(272,380)(273,381)(274,382)(275,383)(276,384)(277,385)(278,386)(279,387)
(280,397)(281,398)(282,399)(283,400)(284,401)(285,402)(286,403)(287,404)
(288,405)(289,388)(290,389)(291,390)(292,391)(293,392)(294,393)(295,394)
(296,395)(297,396)(298,352)(299,353)(300,354)(301,355)(302,356)(303,357)
(304,358)(305,359)(306,360)(307,370)(308,371)(309,372)(310,373)(311,374)
(312,375)(313,376)(314,377)(315,378)(316,361)(317,362)(318,363)(319,364)
(320,365)(321,366)(322,367)(323,368)(324,369);
s1 := Sym(648)!(  1,523)(  2,524)(  3,525)(  4,526)(  5,527)(  6,528)(  7,529)
(  8,530)(  9,531)( 10,514)( 11,515)( 12,516)( 13,517)( 14,518)( 15,519)
( 16,520)( 17,521)( 18,522)( 19,532)( 20,533)( 21,534)( 22,535)( 23,536)
( 24,537)( 25,538)( 26,539)( 27,540)( 28,496)( 29,497)( 30,498)( 31,499)
( 32,500)( 33,501)( 34,502)( 35,503)( 36,504)( 37,487)( 38,488)( 39,489)
( 40,490)( 41,491)( 42,492)( 43,493)( 44,494)( 45,495)( 46,505)( 47,506)
( 48,507)( 49,508)( 50,509)( 51,510)( 52,511)( 53,512)( 54,513)( 55,550)
( 56,551)( 57,552)( 58,553)( 59,554)( 60,555)( 61,556)( 62,557)( 63,558)
( 64,541)( 65,542)( 66,543)( 67,544)( 68,545)( 69,546)( 70,547)( 71,548)
( 72,549)( 73,559)( 74,560)( 75,561)( 76,562)( 77,563)( 78,564)( 79,565)
( 80,566)( 81,567)( 82,604)( 83,605)( 84,606)( 85,607)( 86,608)( 87,609)
( 88,610)( 89,611)( 90,612)( 91,595)( 92,596)( 93,597)( 94,598)( 95,599)
( 96,600)( 97,601)( 98,602)( 99,603)(100,613)(101,614)(102,615)(103,616)
(104,617)(105,618)(106,619)(107,620)(108,621)(109,577)(110,578)(111,579)
(112,580)(113,581)(114,582)(115,583)(116,584)(117,585)(118,568)(119,569)
(120,570)(121,571)(122,572)(123,573)(124,574)(125,575)(126,576)(127,586)
(128,587)(129,588)(130,589)(131,590)(132,591)(133,592)(134,593)(135,594)
(136,631)(137,632)(138,633)(139,634)(140,635)(141,636)(142,637)(143,638)
(144,639)(145,622)(146,623)(147,624)(148,625)(149,626)(150,627)(151,628)
(152,629)(153,630)(154,640)(155,641)(156,642)(157,643)(158,644)(159,645)
(160,646)(161,647)(162,648)(163,442)(164,443)(165,444)(166,445)(167,446)
(168,447)(169,448)(170,449)(171,450)(172,433)(173,434)(174,435)(175,436)
(176,437)(177,438)(178,439)(179,440)(180,441)(181,451)(182,452)(183,453)
(184,454)(185,455)(186,456)(187,457)(188,458)(189,459)(190,415)(191,416)
(192,417)(193,418)(194,419)(195,420)(196,421)(197,422)(198,423)(199,406)
(200,407)(201,408)(202,409)(203,410)(204,411)(205,412)(206,413)(207,414)
(208,424)(209,425)(210,426)(211,427)(212,428)(213,429)(214,430)(215,431)
(216,432)(217,469)(218,470)(219,471)(220,472)(221,473)(222,474)(223,475)
(224,476)(225,477)(226,460)(227,461)(228,462)(229,463)(230,464)(231,465)
(232,466)(233,467)(234,468)(235,478)(236,479)(237,480)(238,481)(239,482)
(240,483)(241,484)(242,485)(243,486)(244,361)(245,362)(246,363)(247,364)
(248,365)(249,366)(250,367)(251,368)(252,369)(253,352)(254,353)(255,354)
(256,355)(257,356)(258,357)(259,358)(260,359)(261,360)(262,370)(263,371)
(264,372)(265,373)(266,374)(267,375)(268,376)(269,377)(270,378)(271,334)
(272,335)(273,336)(274,337)(275,338)(276,339)(277,340)(278,341)(279,342)
(280,325)(281,326)(282,327)(283,328)(284,329)(285,330)(286,331)(287,332)
(288,333)(289,343)(290,344)(291,345)(292,346)(293,347)(294,348)(295,349)
(296,350)(297,351)(298,388)(299,389)(300,390)(301,391)(302,392)(303,393)
(304,394)(305,395)(306,396)(307,379)(308,380)(309,381)(310,382)(311,383)
(312,384)(313,385)(314,386)(315,387)(316,397)(317,398)(318,399)(319,400)
(320,401)(321,402)(322,403)(323,404)(324,405);
s2 := Sym(648)!(  1,325)(  2,327)(  3,326)(  4,333)(  5,332)(  6,331)(  7,330)
(  8,329)(  9,328)( 10,334)( 11,336)( 12,335)( 13,342)( 14,341)( 15,340)
( 16,339)( 17,338)( 18,337)( 19,343)( 20,345)( 21,344)( 22,351)( 23,350)
( 24,349)( 25,348)( 26,347)( 27,346)( 28,379)( 29,381)( 30,380)( 31,387)
( 32,386)( 33,385)( 34,384)( 35,383)( 36,382)( 37,388)( 38,390)( 39,389)
( 40,396)( 41,395)( 42,394)( 43,393)( 44,392)( 45,391)( 46,397)( 47,399)
( 48,398)( 49,405)( 50,404)( 51,403)( 52,402)( 53,401)( 54,400)( 55,352)
( 56,354)( 57,353)( 58,360)( 59,359)( 60,358)( 61,357)( 62,356)( 63,355)
( 64,361)( 65,363)( 66,362)( 67,369)( 68,368)( 69,367)( 70,366)( 71,365)
( 72,364)( 73,370)( 74,372)( 75,371)( 76,378)( 77,377)( 78,376)( 79,375)
( 80,374)( 81,373)( 82,406)( 83,408)( 84,407)( 85,414)( 86,413)( 87,412)
( 88,411)( 89,410)( 90,409)( 91,415)( 92,417)( 93,416)( 94,423)( 95,422)
( 96,421)( 97,420)( 98,419)( 99,418)(100,424)(101,426)(102,425)(103,432)
(104,431)(105,430)(106,429)(107,428)(108,427)(109,460)(110,462)(111,461)
(112,468)(113,467)(114,466)(115,465)(116,464)(117,463)(118,469)(119,471)
(120,470)(121,477)(122,476)(123,475)(124,474)(125,473)(126,472)(127,478)
(128,480)(129,479)(130,486)(131,485)(132,484)(133,483)(134,482)(135,481)
(136,433)(137,435)(138,434)(139,441)(140,440)(141,439)(142,438)(143,437)
(144,436)(145,442)(146,444)(147,443)(148,450)(149,449)(150,448)(151,447)
(152,446)(153,445)(154,451)(155,453)(156,452)(157,459)(158,458)(159,457)
(160,456)(161,455)(162,454)(163,568)(164,570)(165,569)(166,576)(167,575)
(168,574)(169,573)(170,572)(171,571)(172,577)(173,579)(174,578)(175,585)
(176,584)(177,583)(178,582)(179,581)(180,580)(181,586)(182,588)(183,587)
(184,594)(185,593)(186,592)(187,591)(188,590)(189,589)(190,622)(191,624)
(192,623)(193,630)(194,629)(195,628)(196,627)(197,626)(198,625)(199,631)
(200,633)(201,632)(202,639)(203,638)(204,637)(205,636)(206,635)(207,634)
(208,640)(209,642)(210,641)(211,648)(212,647)(213,646)(214,645)(215,644)
(216,643)(217,595)(218,597)(219,596)(220,603)(221,602)(222,601)(223,600)
(224,599)(225,598)(226,604)(227,606)(228,605)(229,612)(230,611)(231,610)
(232,609)(233,608)(234,607)(235,613)(236,615)(237,614)(238,621)(239,620)
(240,619)(241,618)(242,617)(243,616)(244,487)(245,489)(246,488)(247,495)
(248,494)(249,493)(250,492)(251,491)(252,490)(253,496)(254,498)(255,497)
(256,504)(257,503)(258,502)(259,501)(260,500)(261,499)(262,505)(263,507)
(264,506)(265,513)(266,512)(267,511)(268,510)(269,509)(270,508)(271,541)
(272,543)(273,542)(274,549)(275,548)(276,547)(277,546)(278,545)(279,544)
(280,550)(281,552)(282,551)(283,558)(284,557)(285,556)(286,555)(287,554)
(288,553)(289,559)(290,561)(291,560)(292,567)(293,566)(294,565)(295,564)
(296,563)(297,562)(298,514)(299,516)(300,515)(301,522)(302,521)(303,520)
(304,519)(305,518)(306,517)(307,523)(308,525)(309,524)(310,531)(311,530)
(312,529)(313,528)(314,527)(315,526)(316,532)(317,534)(318,533)(319,540)
(320,539)(321,538)(322,537)(323,536)(324,535);
s3 := Sym(648)!(  1,  4)(  2,  6)(  3,  5)(  7,  9)( 10, 13)( 11, 15)( 12, 14)
( 16, 18)( 19, 22)( 20, 24)( 21, 23)( 25, 27)( 28, 31)( 29, 33)( 30, 32)
( 34, 36)( 37, 40)( 38, 42)( 39, 41)( 43, 45)( 46, 49)( 47, 51)( 48, 50)
( 52, 54)( 55, 58)( 56, 60)( 57, 59)( 61, 63)( 64, 67)( 65, 69)( 66, 68)
( 70, 72)( 73, 76)( 74, 78)( 75, 77)( 79, 81)( 82, 85)( 83, 87)( 84, 86)
( 88, 90)( 91, 94)( 92, 96)( 93, 95)( 97, 99)(100,103)(101,105)(102,104)
(106,108)(109,112)(110,114)(111,113)(115,117)(118,121)(119,123)(120,122)
(124,126)(127,130)(128,132)(129,131)(133,135)(136,139)(137,141)(138,140)
(142,144)(145,148)(146,150)(147,149)(151,153)(154,157)(155,159)(156,158)
(160,162)(163,247)(164,249)(165,248)(166,244)(167,246)(168,245)(169,252)
(170,251)(171,250)(172,256)(173,258)(174,257)(175,253)(176,255)(177,254)
(178,261)(179,260)(180,259)(181,265)(182,267)(183,266)(184,262)(185,264)
(186,263)(187,270)(188,269)(189,268)(190,274)(191,276)(192,275)(193,271)
(194,273)(195,272)(196,279)(197,278)(198,277)(199,283)(200,285)(201,284)
(202,280)(203,282)(204,281)(205,288)(206,287)(207,286)(208,292)(209,294)
(210,293)(211,289)(212,291)(213,290)(214,297)(215,296)(216,295)(217,301)
(218,303)(219,302)(220,298)(221,300)(222,299)(223,306)(224,305)(225,304)
(226,310)(227,312)(228,311)(229,307)(230,309)(231,308)(232,315)(233,314)
(234,313)(235,319)(236,321)(237,320)(238,316)(239,318)(240,317)(241,324)
(242,323)(243,322)(325,409)(326,411)(327,410)(328,406)(329,408)(330,407)
(331,414)(332,413)(333,412)(334,418)(335,420)(336,419)(337,415)(338,417)
(339,416)(340,423)(341,422)(342,421)(343,427)(344,429)(345,428)(346,424)
(347,426)(348,425)(349,432)(350,431)(351,430)(352,436)(353,438)(354,437)
(355,433)(356,435)(357,434)(358,441)(359,440)(360,439)(361,445)(362,447)
(363,446)(364,442)(365,444)(366,443)(367,450)(368,449)(369,448)(370,454)
(371,456)(372,455)(373,451)(374,453)(375,452)(376,459)(377,458)(378,457)
(379,463)(380,465)(381,464)(382,460)(383,462)(384,461)(385,468)(386,467)
(387,466)(388,472)(389,474)(390,473)(391,469)(392,471)(393,470)(394,477)
(395,476)(396,475)(397,481)(398,483)(399,482)(400,478)(401,480)(402,479)
(403,486)(404,485)(405,484)(487,490)(488,492)(489,491)(493,495)(496,499)
(497,501)(498,500)(502,504)(505,508)(506,510)(507,509)(511,513)(514,517)
(515,519)(516,518)(520,522)(523,526)(524,528)(525,527)(529,531)(532,535)
(533,537)(534,536)(538,540)(541,544)(542,546)(543,545)(547,549)(550,553)
(551,555)(552,554)(556,558)(559,562)(560,564)(561,563)(565,567)(568,571)
(569,573)(570,572)(574,576)(577,580)(578,582)(579,581)(583,585)(586,589)
(587,591)(588,590)(592,594)(595,598)(596,600)(597,599)(601,603)(604,607)
(605,609)(606,608)(610,612)(613,616)(614,618)(615,617)(619,621)(622,625)
(623,627)(624,626)(628,630)(631,634)(632,636)(633,635)(637,639)(640,643)
(641,645)(642,644)(646,648);
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, 
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*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*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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