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Polytope of Type {3,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,12}*1296d
if this polytope has a name.
Group : SmallGroup(1296,2061)
Rank : 4
Schlafli Type : {3,6,12}
Number of vertices, edges, etc : 3, 27, 108, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 6
Special Properties :
   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 : {3,6,6}*648d
   3-fold quotients : {3,6,12}*432b
   6-fold quotients : {3,6,6}*216b
   9-fold quotients : {3,2,12}*144, {3,6,4}*144
   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)(  2,489)(  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)( 28,541)( 29,543)( 30,542)( 31,544)( 32,546)
( 33,545)( 34,547)( 35,549)( 36,548)( 37,559)( 38,561)( 39,560)( 40,562)
( 41,564)( 42,563)( 43,565)( 44,567)( 45,566)( 46,550)( 47,552)( 48,551)
( 49,553)( 50,555)( 51,554)( 52,556)( 53,558)( 54,557)( 55,514)( 56,516)
( 57,515)( 58,517)( 59,519)( 60,518)( 61,520)( 62,522)( 63,521)( 64,532)
( 65,534)( 66,533)( 67,535)( 68,537)( 69,536)( 70,538)( 71,540)( 72,539)
( 73,523)( 74,525)( 75,524)( 76,526)( 77,528)( 78,527)( 79,529)( 80,531)
( 81,530)( 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)(109,622)(110,624)(111,623)(112,625)
(113,627)(114,626)(115,628)(116,630)(117,629)(118,640)(119,642)(120,641)
(121,643)(122,645)(123,644)(124,646)(125,648)(126,647)(127,631)(128,633)
(129,632)(130,634)(131,636)(132,635)(133,637)(134,639)(135,638)(136,595)
(137,597)(138,596)(139,598)(140,600)(141,599)(142,601)(143,603)(144,602)
(145,613)(146,615)(147,614)(148,616)(149,618)(150,617)(151,619)(152,621)
(153,620)(154,604)(155,606)(156,605)(157,607)(158,609)(159,608)(160,610)
(161,612)(162,611)(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)(190,460)(191,462)(192,461)
(193,463)(194,465)(195,464)(196,466)(197,468)(198,467)(199,478)(200,480)
(201,479)(202,481)(203,483)(204,482)(205,484)(206,486)(207,485)(208,469)
(209,471)(210,470)(211,472)(212,474)(213,473)(214,475)(215,477)(216,476)
(217,433)(218,435)(219,434)(220,436)(221,438)(222,437)(223,439)(224,441)
(225,440)(226,451)(227,453)(228,452)(229,454)(230,456)(231,455)(232,457)
(233,459)(234,458)(235,442)(236,444)(237,443)(238,445)(239,447)(240,446)
(241,448)(242,450)(243,449)(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)(271,379)(272,381)
(273,380)(274,382)(275,384)(276,383)(277,385)(278,387)(279,386)(280,397)
(281,399)(282,398)(283,400)(284,402)(285,401)(286,403)(287,405)(288,404)
(289,388)(290,390)(291,389)(292,391)(293,393)(294,392)(295,394)(296,396)
(297,395)(298,352)(299,354)(300,353)(301,355)(302,357)(303,356)(304,358)
(305,360)(306,359)(307,370)(308,372)(309,371)(310,373)(311,375)(312,374)
(313,376)(314,378)(315,377)(316,361)(317,363)(318,362)(319,364)(320,366)
(321,365)(322,367)(323,369)(324,368);;
s1 := (  1,523)(  2,525)(  3,524)(  4,527)(  5,526)(  6,528)(  7,531)(  8,530)
(  9,529)( 10,514)( 11,516)( 12,515)( 13,518)( 14,517)( 15,519)( 16,522)
( 17,521)( 18,520)( 19,532)( 20,534)( 21,533)( 22,536)( 23,535)( 24,537)
( 25,540)( 26,539)( 27,538)( 28,496)( 29,498)( 30,497)( 31,500)( 32,499)
( 33,501)( 34,504)( 35,503)( 36,502)( 37,487)( 38,489)( 39,488)( 40,491)
( 41,490)( 42,492)( 43,495)( 44,494)( 45,493)( 46,505)( 47,507)( 48,506)
( 49,509)( 50,508)( 51,510)( 52,513)( 53,512)( 54,511)( 55,550)( 56,552)
( 57,551)( 58,554)( 59,553)( 60,555)( 61,558)( 62,557)( 63,556)( 64,541)
( 65,543)( 66,542)( 67,545)( 68,544)( 69,546)( 70,549)( 71,548)( 72,547)
( 73,559)( 74,561)( 75,560)( 76,563)( 77,562)( 78,564)( 79,567)( 80,566)
( 81,565)( 82,604)( 83,606)( 84,605)( 85,608)( 86,607)( 87,609)( 88,612)
( 89,611)( 90,610)( 91,595)( 92,597)( 93,596)( 94,599)( 95,598)( 96,600)
( 97,603)( 98,602)( 99,601)(100,613)(101,615)(102,614)(103,617)(104,616)
(105,618)(106,621)(107,620)(108,619)(109,577)(110,579)(111,578)(112,581)
(113,580)(114,582)(115,585)(116,584)(117,583)(118,568)(119,570)(120,569)
(121,572)(122,571)(123,573)(124,576)(125,575)(126,574)(127,586)(128,588)
(129,587)(130,590)(131,589)(132,591)(133,594)(134,593)(135,592)(136,631)
(137,633)(138,632)(139,635)(140,634)(141,636)(142,639)(143,638)(144,637)
(145,622)(146,624)(147,623)(148,626)(149,625)(150,627)(151,630)(152,629)
(153,628)(154,640)(155,642)(156,641)(157,644)(158,643)(159,645)(160,648)
(161,647)(162,646)(163,442)(164,444)(165,443)(166,446)(167,445)(168,447)
(169,450)(170,449)(171,448)(172,433)(173,435)(174,434)(175,437)(176,436)
(177,438)(178,441)(179,440)(180,439)(181,451)(182,453)(183,452)(184,455)
(185,454)(186,456)(187,459)(188,458)(189,457)(190,415)(191,417)(192,416)
(193,419)(194,418)(195,420)(196,423)(197,422)(198,421)(199,406)(200,408)
(201,407)(202,410)(203,409)(204,411)(205,414)(206,413)(207,412)(208,424)
(209,426)(210,425)(211,428)(212,427)(213,429)(214,432)(215,431)(216,430)
(217,469)(218,471)(219,470)(220,473)(221,472)(222,474)(223,477)(224,476)
(225,475)(226,460)(227,462)(228,461)(229,464)(230,463)(231,465)(232,468)
(233,467)(234,466)(235,478)(236,480)(237,479)(238,482)(239,481)(240,483)
(241,486)(242,485)(243,484)(244,361)(245,363)(246,362)(247,365)(248,364)
(249,366)(250,369)(251,368)(252,367)(253,352)(254,354)(255,353)(256,356)
(257,355)(258,357)(259,360)(260,359)(261,358)(262,370)(263,372)(264,371)
(265,374)(266,373)(267,375)(268,378)(269,377)(270,376)(271,334)(272,336)
(273,335)(274,338)(275,337)(276,339)(277,342)(278,341)(279,340)(280,325)
(281,327)(282,326)(283,329)(284,328)(285,330)(286,333)(287,332)(288,331)
(289,343)(290,345)(291,344)(292,347)(293,346)(294,348)(295,351)(296,350)
(297,349)(298,388)(299,390)(300,389)(301,392)(302,391)(303,393)(304,396)
(305,395)(306,394)(307,379)(308,381)(309,380)(310,383)(311,382)(312,384)
(313,387)(314,386)(315,385)(316,397)(317,399)(318,398)(319,401)(320,400)
(321,402)(322,405)(323,404)(324,403);;
s2 := (  1,328)(  2,329)(  3,330)(  4,325)(  5,326)(  6,327)(  7,331)(  8,332)
(  9,333)( 10,337)( 11,338)( 12,339)( 13,334)( 14,335)( 15,336)( 16,340)
( 17,341)( 18,342)( 19,346)( 20,347)( 21,348)( 22,343)( 23,344)( 24,345)
( 25,349)( 26,350)( 27,351)( 28,382)( 29,383)( 30,384)( 31,379)( 32,380)
( 33,381)( 34,385)( 35,386)( 36,387)( 37,391)( 38,392)( 39,393)( 40,388)
( 41,389)( 42,390)( 43,394)( 44,395)( 45,396)( 46,400)( 47,401)( 48,402)
( 49,397)( 50,398)( 51,399)( 52,403)( 53,404)( 54,405)( 55,355)( 56,356)
( 57,357)( 58,352)( 59,353)( 60,354)( 61,358)( 62,359)( 63,360)( 64,364)
( 65,365)( 66,366)( 67,361)( 68,362)( 69,363)( 70,367)( 71,368)( 72,369)
( 73,373)( 74,374)( 75,375)( 76,370)( 77,371)( 78,372)( 79,376)( 80,377)
( 81,378)( 82,409)( 83,410)( 84,411)( 85,406)( 86,407)( 87,408)( 88,412)
( 89,413)( 90,414)( 91,418)( 92,419)( 93,420)( 94,415)( 95,416)( 96,417)
( 97,421)( 98,422)( 99,423)(100,427)(101,428)(102,429)(103,424)(104,425)
(105,426)(106,430)(107,431)(108,432)(109,463)(110,464)(111,465)(112,460)
(113,461)(114,462)(115,466)(116,467)(117,468)(118,472)(119,473)(120,474)
(121,469)(122,470)(123,471)(124,475)(125,476)(126,477)(127,481)(128,482)
(129,483)(130,478)(131,479)(132,480)(133,484)(134,485)(135,486)(136,436)
(137,437)(138,438)(139,433)(140,434)(141,435)(142,439)(143,440)(144,441)
(145,445)(146,446)(147,447)(148,442)(149,443)(150,444)(151,448)(152,449)
(153,450)(154,454)(155,455)(156,456)(157,451)(158,452)(159,453)(160,457)
(161,458)(162,459)(163,571)(164,572)(165,573)(166,568)(167,569)(168,570)
(169,574)(170,575)(171,576)(172,580)(173,581)(174,582)(175,577)(176,578)
(177,579)(178,583)(179,584)(180,585)(181,589)(182,590)(183,591)(184,586)
(185,587)(186,588)(187,592)(188,593)(189,594)(190,625)(191,626)(192,627)
(193,622)(194,623)(195,624)(196,628)(197,629)(198,630)(199,634)(200,635)
(201,636)(202,631)(203,632)(204,633)(205,637)(206,638)(207,639)(208,643)
(209,644)(210,645)(211,640)(212,641)(213,642)(214,646)(215,647)(216,648)
(217,598)(218,599)(219,600)(220,595)(221,596)(222,597)(223,601)(224,602)
(225,603)(226,607)(227,608)(228,609)(229,604)(230,605)(231,606)(232,610)
(233,611)(234,612)(235,616)(236,617)(237,618)(238,613)(239,614)(240,615)
(241,619)(242,620)(243,621)(244,490)(245,491)(246,492)(247,487)(248,488)
(249,489)(250,493)(251,494)(252,495)(253,499)(254,500)(255,501)(256,496)
(257,497)(258,498)(259,502)(260,503)(261,504)(262,508)(263,509)(264,510)
(265,505)(266,506)(267,507)(268,511)(269,512)(270,513)(271,544)(272,545)
(273,546)(274,541)(275,542)(276,543)(277,547)(278,548)(279,549)(280,553)
(281,554)(282,555)(283,550)(284,551)(285,552)(286,556)(287,557)(288,558)
(289,562)(290,563)(291,564)(292,559)(293,560)(294,561)(295,565)(296,566)
(297,567)(298,517)(299,518)(300,519)(301,514)(302,515)(303,516)(304,520)
(305,521)(306,522)(307,526)(308,527)(309,528)(310,523)(311,524)(312,525)
(313,529)(314,530)(315,531)(316,535)(317,536)(318,537)(319,532)(320,533)
(321,534)(322,538)(323,539)(324,540);;
s3 := (  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)
(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)
(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)
(163,244)(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)
(171,248)(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)
(179,258)(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)
(187,265)(188,267)(189,266)(190,271)(191,273)(192,272)(193,277)(194,279)
(195,278)(196,274)(197,276)(198,275)(199,280)(200,282)(201,281)(202,286)
(203,288)(204,287)(205,283)(206,285)(207,284)(208,289)(209,291)(210,290)
(211,295)(212,297)(213,296)(214,292)(215,294)(216,293)(217,298)(218,300)
(219,299)(220,304)(221,306)(222,305)(223,301)(224,303)(225,302)(226,307)
(227,309)(228,308)(229,313)(230,315)(231,314)(232,310)(233,312)(234,311)
(235,316)(236,318)(237,317)(238,322)(239,324)(240,323)(241,319)(242,321)
(243,320)(325,406)(326,408)(327,407)(328,412)(329,414)(330,413)(331,409)
(332,411)(333,410)(334,415)(335,417)(336,416)(337,421)(338,423)(339,422)
(340,418)(341,420)(342,419)(343,424)(344,426)(345,425)(346,430)(347,432)
(348,431)(349,427)(350,429)(351,428)(352,433)(353,435)(354,434)(355,439)
(356,441)(357,440)(358,436)(359,438)(360,437)(361,442)(362,444)(363,443)
(364,448)(365,450)(366,449)(367,445)(368,447)(369,446)(370,451)(371,453)
(372,452)(373,457)(374,459)(375,458)(376,454)(377,456)(378,455)(379,460)
(380,462)(381,461)(382,466)(383,468)(384,467)(385,463)(386,465)(387,464)
(388,469)(389,471)(390,470)(391,475)(392,477)(393,476)(394,472)(395,474)
(396,473)(397,478)(398,480)(399,479)(400,484)(401,486)(402,485)(403,481)
(404,483)(405,482)(488,489)(490,493)(491,495)(492,494)(497,498)(499,502)
(500,504)(501,503)(506,507)(508,511)(509,513)(510,512)(515,516)(517,520)
(518,522)(519,521)(524,525)(526,529)(527,531)(528,530)(533,534)(535,538)
(536,540)(537,539)(542,543)(544,547)(545,549)(546,548)(551,552)(553,556)
(554,558)(555,557)(560,561)(562,565)(563,567)(564,566)(569,570)(571,574)
(572,576)(573,575)(578,579)(580,583)(581,585)(582,584)(587,588)(589,592)
(590,594)(591,593)(596,597)(598,601)(599,603)(600,602)(605,606)(607,610)
(608,612)(609,611)(614,615)(616,619)(617,621)(618,620)(623,624)(625,628)
(626,630)(627,629)(632,633)(634,637)(635,639)(636,638)(641,642)(643,646)
(644,648)(645,647);;
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, 
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s0*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*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 ];;
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)( 28,541)( 29,543)( 30,542)( 31,544)
( 32,546)( 33,545)( 34,547)( 35,549)( 36,548)( 37,559)( 38,561)( 39,560)
( 40,562)( 41,564)( 42,563)( 43,565)( 44,567)( 45,566)( 46,550)( 47,552)
( 48,551)( 49,553)( 50,555)( 51,554)( 52,556)( 53,558)( 54,557)( 55,514)
( 56,516)( 57,515)( 58,517)( 59,519)( 60,518)( 61,520)( 62,522)( 63,521)
( 64,532)( 65,534)( 66,533)( 67,535)( 68,537)( 69,536)( 70,538)( 71,540)
( 72,539)( 73,523)( 74,525)( 75,524)( 76,526)( 77,528)( 78,527)( 79,529)
( 80,531)( 81,530)( 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)(109,622)(110,624)(111,623)
(112,625)(113,627)(114,626)(115,628)(116,630)(117,629)(118,640)(119,642)
(120,641)(121,643)(122,645)(123,644)(124,646)(125,648)(126,647)(127,631)
(128,633)(129,632)(130,634)(131,636)(132,635)(133,637)(134,639)(135,638)
(136,595)(137,597)(138,596)(139,598)(140,600)(141,599)(142,601)(143,603)
(144,602)(145,613)(146,615)(147,614)(148,616)(149,618)(150,617)(151,619)
(152,621)(153,620)(154,604)(155,606)(156,605)(157,607)(158,609)(159,608)
(160,610)(161,612)(162,611)(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)(190,460)(191,462)
(192,461)(193,463)(194,465)(195,464)(196,466)(197,468)(198,467)(199,478)
(200,480)(201,479)(202,481)(203,483)(204,482)(205,484)(206,486)(207,485)
(208,469)(209,471)(210,470)(211,472)(212,474)(213,473)(214,475)(215,477)
(216,476)(217,433)(218,435)(219,434)(220,436)(221,438)(222,437)(223,439)
(224,441)(225,440)(226,451)(227,453)(228,452)(229,454)(230,456)(231,455)
(232,457)(233,459)(234,458)(235,442)(236,444)(237,443)(238,445)(239,447)
(240,446)(241,448)(242,450)(243,449)(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)(271,379)
(272,381)(273,380)(274,382)(275,384)(276,383)(277,385)(278,387)(279,386)
(280,397)(281,399)(282,398)(283,400)(284,402)(285,401)(286,403)(287,405)
(288,404)(289,388)(290,390)(291,389)(292,391)(293,393)(294,392)(295,394)
(296,396)(297,395)(298,352)(299,354)(300,353)(301,355)(302,357)(303,356)
(304,358)(305,360)(306,359)(307,370)(308,372)(309,371)(310,373)(311,375)
(312,374)(313,376)(314,378)(315,377)(316,361)(317,363)(318,362)(319,364)
(320,366)(321,365)(322,367)(323,369)(324,368);
s1 := Sym(648)!(  1,523)(  2,525)(  3,524)(  4,527)(  5,526)(  6,528)(  7,531)
(  8,530)(  9,529)( 10,514)( 11,516)( 12,515)( 13,518)( 14,517)( 15,519)
( 16,522)( 17,521)( 18,520)( 19,532)( 20,534)( 21,533)( 22,536)( 23,535)
( 24,537)( 25,540)( 26,539)( 27,538)( 28,496)( 29,498)( 30,497)( 31,500)
( 32,499)( 33,501)( 34,504)( 35,503)( 36,502)( 37,487)( 38,489)( 39,488)
( 40,491)( 41,490)( 42,492)( 43,495)( 44,494)( 45,493)( 46,505)( 47,507)
( 48,506)( 49,509)( 50,508)( 51,510)( 52,513)( 53,512)( 54,511)( 55,550)
( 56,552)( 57,551)( 58,554)( 59,553)( 60,555)( 61,558)( 62,557)( 63,556)
( 64,541)( 65,543)( 66,542)( 67,545)( 68,544)( 69,546)( 70,549)( 71,548)
( 72,547)( 73,559)( 74,561)( 75,560)( 76,563)( 77,562)( 78,564)( 79,567)
( 80,566)( 81,565)( 82,604)( 83,606)( 84,605)( 85,608)( 86,607)( 87,609)
( 88,612)( 89,611)( 90,610)( 91,595)( 92,597)( 93,596)( 94,599)( 95,598)
( 96,600)( 97,603)( 98,602)( 99,601)(100,613)(101,615)(102,614)(103,617)
(104,616)(105,618)(106,621)(107,620)(108,619)(109,577)(110,579)(111,578)
(112,581)(113,580)(114,582)(115,585)(116,584)(117,583)(118,568)(119,570)
(120,569)(121,572)(122,571)(123,573)(124,576)(125,575)(126,574)(127,586)
(128,588)(129,587)(130,590)(131,589)(132,591)(133,594)(134,593)(135,592)
(136,631)(137,633)(138,632)(139,635)(140,634)(141,636)(142,639)(143,638)
(144,637)(145,622)(146,624)(147,623)(148,626)(149,625)(150,627)(151,630)
(152,629)(153,628)(154,640)(155,642)(156,641)(157,644)(158,643)(159,645)
(160,648)(161,647)(162,646)(163,442)(164,444)(165,443)(166,446)(167,445)
(168,447)(169,450)(170,449)(171,448)(172,433)(173,435)(174,434)(175,437)
(176,436)(177,438)(178,441)(179,440)(180,439)(181,451)(182,453)(183,452)
(184,455)(185,454)(186,456)(187,459)(188,458)(189,457)(190,415)(191,417)
(192,416)(193,419)(194,418)(195,420)(196,423)(197,422)(198,421)(199,406)
(200,408)(201,407)(202,410)(203,409)(204,411)(205,414)(206,413)(207,412)
(208,424)(209,426)(210,425)(211,428)(212,427)(213,429)(214,432)(215,431)
(216,430)(217,469)(218,471)(219,470)(220,473)(221,472)(222,474)(223,477)
(224,476)(225,475)(226,460)(227,462)(228,461)(229,464)(230,463)(231,465)
(232,468)(233,467)(234,466)(235,478)(236,480)(237,479)(238,482)(239,481)
(240,483)(241,486)(242,485)(243,484)(244,361)(245,363)(246,362)(247,365)
(248,364)(249,366)(250,369)(251,368)(252,367)(253,352)(254,354)(255,353)
(256,356)(257,355)(258,357)(259,360)(260,359)(261,358)(262,370)(263,372)
(264,371)(265,374)(266,373)(267,375)(268,378)(269,377)(270,376)(271,334)
(272,336)(273,335)(274,338)(275,337)(276,339)(277,342)(278,341)(279,340)
(280,325)(281,327)(282,326)(283,329)(284,328)(285,330)(286,333)(287,332)
(288,331)(289,343)(290,345)(291,344)(292,347)(293,346)(294,348)(295,351)
(296,350)(297,349)(298,388)(299,390)(300,389)(301,392)(302,391)(303,393)
(304,396)(305,395)(306,394)(307,379)(308,381)(309,380)(310,383)(311,382)
(312,384)(313,387)(314,386)(315,385)(316,397)(317,399)(318,398)(319,401)
(320,400)(321,402)(322,405)(323,404)(324,403);
s2 := Sym(648)!(  1,328)(  2,329)(  3,330)(  4,325)(  5,326)(  6,327)(  7,331)
(  8,332)(  9,333)( 10,337)( 11,338)( 12,339)( 13,334)( 14,335)( 15,336)
( 16,340)( 17,341)( 18,342)( 19,346)( 20,347)( 21,348)( 22,343)( 23,344)
( 24,345)( 25,349)( 26,350)( 27,351)( 28,382)( 29,383)( 30,384)( 31,379)
( 32,380)( 33,381)( 34,385)( 35,386)( 36,387)( 37,391)( 38,392)( 39,393)
( 40,388)( 41,389)( 42,390)( 43,394)( 44,395)( 45,396)( 46,400)( 47,401)
( 48,402)( 49,397)( 50,398)( 51,399)( 52,403)( 53,404)( 54,405)( 55,355)
( 56,356)( 57,357)( 58,352)( 59,353)( 60,354)( 61,358)( 62,359)( 63,360)
( 64,364)( 65,365)( 66,366)( 67,361)( 68,362)( 69,363)( 70,367)( 71,368)
( 72,369)( 73,373)( 74,374)( 75,375)( 76,370)( 77,371)( 78,372)( 79,376)
( 80,377)( 81,378)( 82,409)( 83,410)( 84,411)( 85,406)( 86,407)( 87,408)
( 88,412)( 89,413)( 90,414)( 91,418)( 92,419)( 93,420)( 94,415)( 95,416)
( 96,417)( 97,421)( 98,422)( 99,423)(100,427)(101,428)(102,429)(103,424)
(104,425)(105,426)(106,430)(107,431)(108,432)(109,463)(110,464)(111,465)
(112,460)(113,461)(114,462)(115,466)(116,467)(117,468)(118,472)(119,473)
(120,474)(121,469)(122,470)(123,471)(124,475)(125,476)(126,477)(127,481)
(128,482)(129,483)(130,478)(131,479)(132,480)(133,484)(134,485)(135,486)
(136,436)(137,437)(138,438)(139,433)(140,434)(141,435)(142,439)(143,440)
(144,441)(145,445)(146,446)(147,447)(148,442)(149,443)(150,444)(151,448)
(152,449)(153,450)(154,454)(155,455)(156,456)(157,451)(158,452)(159,453)
(160,457)(161,458)(162,459)(163,571)(164,572)(165,573)(166,568)(167,569)
(168,570)(169,574)(170,575)(171,576)(172,580)(173,581)(174,582)(175,577)
(176,578)(177,579)(178,583)(179,584)(180,585)(181,589)(182,590)(183,591)
(184,586)(185,587)(186,588)(187,592)(188,593)(189,594)(190,625)(191,626)
(192,627)(193,622)(194,623)(195,624)(196,628)(197,629)(198,630)(199,634)
(200,635)(201,636)(202,631)(203,632)(204,633)(205,637)(206,638)(207,639)
(208,643)(209,644)(210,645)(211,640)(212,641)(213,642)(214,646)(215,647)
(216,648)(217,598)(218,599)(219,600)(220,595)(221,596)(222,597)(223,601)
(224,602)(225,603)(226,607)(227,608)(228,609)(229,604)(230,605)(231,606)
(232,610)(233,611)(234,612)(235,616)(236,617)(237,618)(238,613)(239,614)
(240,615)(241,619)(242,620)(243,621)(244,490)(245,491)(246,492)(247,487)
(248,488)(249,489)(250,493)(251,494)(252,495)(253,499)(254,500)(255,501)
(256,496)(257,497)(258,498)(259,502)(260,503)(261,504)(262,508)(263,509)
(264,510)(265,505)(266,506)(267,507)(268,511)(269,512)(270,513)(271,544)
(272,545)(273,546)(274,541)(275,542)(276,543)(277,547)(278,548)(279,549)
(280,553)(281,554)(282,555)(283,550)(284,551)(285,552)(286,556)(287,557)
(288,558)(289,562)(290,563)(291,564)(292,559)(293,560)(294,561)(295,565)
(296,566)(297,567)(298,517)(299,518)(300,519)(301,514)(302,515)(303,516)
(304,520)(305,521)(306,522)(307,526)(308,527)(309,528)(310,523)(311,524)
(312,525)(313,529)(314,530)(315,531)(316,535)(317,536)(318,537)(319,532)
(320,533)(321,534)(322,538)(323,539)(324,540);
s3 := Sym(648)!(  2,  3)(  4,  7)(  5,  9)(  6,  8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)
(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)
(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)
(159,161)(163,244)(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)
(170,249)(171,248)(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)
(178,256)(179,258)(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)
(186,269)(187,265)(188,267)(189,266)(190,271)(191,273)(192,272)(193,277)
(194,279)(195,278)(196,274)(197,276)(198,275)(199,280)(200,282)(201,281)
(202,286)(203,288)(204,287)(205,283)(206,285)(207,284)(208,289)(209,291)
(210,290)(211,295)(212,297)(213,296)(214,292)(215,294)(216,293)(217,298)
(218,300)(219,299)(220,304)(221,306)(222,305)(223,301)(224,303)(225,302)
(226,307)(227,309)(228,308)(229,313)(230,315)(231,314)(232,310)(233,312)
(234,311)(235,316)(236,318)(237,317)(238,322)(239,324)(240,323)(241,319)
(242,321)(243,320)(325,406)(326,408)(327,407)(328,412)(329,414)(330,413)
(331,409)(332,411)(333,410)(334,415)(335,417)(336,416)(337,421)(338,423)
(339,422)(340,418)(341,420)(342,419)(343,424)(344,426)(345,425)(346,430)
(347,432)(348,431)(349,427)(350,429)(351,428)(352,433)(353,435)(354,434)
(355,439)(356,441)(357,440)(358,436)(359,438)(360,437)(361,442)(362,444)
(363,443)(364,448)(365,450)(366,449)(367,445)(368,447)(369,446)(370,451)
(371,453)(372,452)(373,457)(374,459)(375,458)(376,454)(377,456)(378,455)
(379,460)(380,462)(381,461)(382,466)(383,468)(384,467)(385,463)(386,465)
(387,464)(388,469)(389,471)(390,470)(391,475)(392,477)(393,476)(394,472)
(395,474)(396,473)(397,478)(398,480)(399,479)(400,484)(401,486)(402,485)
(403,481)(404,483)(405,482)(488,489)(490,493)(491,495)(492,494)(497,498)
(499,502)(500,504)(501,503)(506,507)(508,511)(509,513)(510,512)(515,516)
(517,520)(518,522)(519,521)(524,525)(526,529)(527,531)(528,530)(533,534)
(535,538)(536,540)(537,539)(542,543)(544,547)(545,549)(546,548)(551,552)
(553,556)(554,558)(555,557)(560,561)(562,565)(563,567)(564,566)(569,570)
(571,574)(572,576)(573,575)(578,579)(580,583)(581,585)(582,584)(587,588)
(589,592)(590,594)(591,593)(596,597)(598,601)(599,603)(600,602)(605,606)
(607,610)(608,612)(609,611)(614,615)(616,619)(617,621)(618,620)(623,624)
(625,628)(626,630)(627,629)(632,633)(634,637)(635,639)(636,638)(641,642)
(643,646)(644,648)(645,647);
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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, 
s3*s0*s2*s1*s2*s3*s2*s3*s2*s1*s2*s3*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 >; 
 
References : None.
to this polytope