Polytope of Type {3,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,12}*1296e
Also Known As : {{3,6}6,{6,12|2}}. if this polytope has another name.
Group : SmallGroup(1296,2077)
Rank : 4
Schlafli Type : {3,6,12}
Number of vertices, edges, etc : 9, 27, 108, 12
Order of s0s1s2s3 : 12
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 : {3,6,6}*648c
3-fold quotients : {3,6,4}*432a, {3,6,12}*432b
6-fold quotients : {3,6,2}*216, {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)( 2,488)( 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 := ( 1,551)( 2,552)( 3,550)( 4,554)( 5,555)( 6,553)( 7,557)( 8,558)
( 9,556)( 10,543)( 11,541)( 12,542)( 13,546)( 14,544)( 15,545)( 16,549)
( 17,547)( 18,548)( 19,559)( 20,560)( 21,561)( 22,562)( 23,563)( 24,564)
( 25,565)( 26,566)( 27,567)( 28,524)( 29,525)( 30,523)( 31,527)( 32,528)
( 33,526)( 34,530)( 35,531)( 36,529)( 37,516)( 38,514)( 39,515)( 40,519)
( 41,517)( 42,518)( 43,522)( 44,520)( 45,521)( 46,532)( 47,533)( 48,534)
( 49,535)( 50,536)( 51,537)( 52,538)( 53,539)( 54,540)( 55,497)( 56,498)
( 57,496)( 58,500)( 59,501)( 60,499)( 61,503)( 62,504)( 63,502)( 64,489)
( 65,487)( 66,488)( 67,492)( 68,490)( 69,491)( 70,495)( 71,493)( 72,494)
( 73,505)( 74,506)( 75,507)( 76,508)( 77,509)( 78,510)( 79,511)( 80,512)
( 81,513)( 82,632)( 83,633)( 84,631)( 85,635)( 86,636)( 87,634)( 88,638)
( 89,639)( 90,637)( 91,624)( 92,622)( 93,623)( 94,627)( 95,625)( 96,626)
( 97,630)( 98,628)( 99,629)(100,640)(101,641)(102,642)(103,643)(104,644)
(105,645)(106,646)(107,647)(108,648)(109,605)(110,606)(111,604)(112,608)
(113,609)(114,607)(115,611)(116,612)(117,610)(118,597)(119,595)(120,596)
(121,600)(122,598)(123,599)(124,603)(125,601)(126,602)(127,613)(128,614)
(129,615)(130,616)(131,617)(132,618)(133,619)(134,620)(135,621)(136,578)
(137,579)(138,577)(139,581)(140,582)(141,580)(142,584)(143,585)(144,583)
(145,570)(146,568)(147,569)(148,573)(149,571)(150,572)(151,576)(152,574)
(153,575)(154,586)(155,587)(156,588)(157,589)(158,590)(159,591)(160,592)
(161,593)(162,594)(163,470)(164,471)(165,469)(166,473)(167,474)(168,472)
(169,476)(170,477)(171,475)(172,462)(173,460)(174,461)(175,465)(176,463)
(177,464)(178,468)(179,466)(180,467)(181,478)(182,479)(183,480)(184,481)
(185,482)(186,483)(187,484)(188,485)(189,486)(190,443)(191,444)(192,442)
(193,446)(194,447)(195,445)(196,449)(197,450)(198,448)(199,435)(200,433)
(201,434)(202,438)(203,436)(204,437)(205,441)(206,439)(207,440)(208,451)
(209,452)(210,453)(211,454)(212,455)(213,456)(214,457)(215,458)(216,459)
(217,416)(218,417)(219,415)(220,419)(221,420)(222,418)(223,422)(224,423)
(225,421)(226,408)(227,406)(228,407)(229,411)(230,409)(231,410)(232,414)
(233,412)(234,413)(235,424)(236,425)(237,426)(238,427)(239,428)(240,429)
(241,430)(242,431)(243,432)(244,389)(245,390)(246,388)(247,392)(248,393)
(249,391)(250,395)(251,396)(252,394)(253,381)(254,379)(255,380)(256,384)
(257,382)(258,383)(259,387)(260,385)(261,386)(262,397)(263,398)(264,399)
(265,400)(266,401)(267,402)(268,403)(269,404)(270,405)(271,362)(272,363)
(273,361)(274,365)(275,366)(276,364)(277,368)(278,369)(279,367)(280,354)
(281,352)(282,353)(283,357)(284,355)(285,356)(286,360)(287,358)(288,359)
(289,370)(290,371)(291,372)(292,373)(293,374)(294,375)(295,376)(296,377)
(297,378)(298,335)(299,336)(300,334)(301,338)(302,339)(303,337)(304,341)
(305,342)(306,340)(307,327)(308,325)(309,326)(310,330)(311,328)(312,329)
(313,333)(314,331)(315,332)(316,343)(317,344)(318,345)(319,346)(320,347)
(321,348)(322,349)(323,350)(324,351);;
s2 := ( 1,325)( 2,327)( 3,326)( 4,331)( 5,333)( 6,332)( 7,328)( 8,330)
( 9,329)( 10,334)( 11,336)( 12,335)( 13,340)( 14,342)( 15,341)( 16,337)
( 17,339)( 18,338)( 19,343)( 20,345)( 21,344)( 22,349)( 23,351)( 24,350)
( 25,346)( 26,348)( 27,347)( 28,379)( 29,381)( 30,380)( 31,385)( 32,387)
( 33,386)( 34,382)( 35,384)( 36,383)( 37,388)( 38,390)( 39,389)( 40,394)
( 41,396)( 42,395)( 43,391)( 44,393)( 45,392)( 46,397)( 47,399)( 48,398)
( 49,403)( 50,405)( 51,404)( 52,400)( 53,402)( 54,401)( 55,352)( 56,354)
( 57,353)( 58,358)( 59,360)( 60,359)( 61,355)( 62,357)( 63,356)( 64,361)
( 65,363)( 66,362)( 67,367)( 68,369)( 69,368)( 70,364)( 71,366)( 72,365)
( 73,370)( 74,372)( 75,371)( 76,376)( 77,378)( 78,377)( 79,373)( 80,375)
( 81,374)( 82,406)( 83,408)( 84,407)( 85,412)( 86,414)( 87,413)( 88,409)
( 89,411)( 90,410)( 91,415)( 92,417)( 93,416)( 94,421)( 95,423)( 96,422)
( 97,418)( 98,420)( 99,419)(100,424)(101,426)(102,425)(103,430)(104,432)
(105,431)(106,427)(107,429)(108,428)(109,460)(110,462)(111,461)(112,466)
(113,468)(114,467)(115,463)(116,465)(117,464)(118,469)(119,471)(120,470)
(121,475)(122,477)(123,476)(124,472)(125,474)(126,473)(127,478)(128,480)
(129,479)(130,484)(131,486)(132,485)(133,481)(134,483)(135,482)(136,433)
(137,435)(138,434)(139,439)(140,441)(141,440)(142,436)(143,438)(144,437)
(145,442)(146,444)(147,443)(148,448)(149,450)(150,449)(151,445)(152,447)
(153,446)(154,451)(155,453)(156,452)(157,457)(158,459)(159,458)(160,454)
(161,456)(162,455)(163,568)(164,570)(165,569)(166,574)(167,576)(168,575)
(169,571)(170,573)(171,572)(172,577)(173,579)(174,578)(175,583)(176,585)
(177,584)(178,580)(179,582)(180,581)(181,586)(182,588)(183,587)(184,592)
(185,594)(186,593)(187,589)(188,591)(189,590)(190,622)(191,624)(192,623)
(193,628)(194,630)(195,629)(196,625)(197,627)(198,626)(199,631)(200,633)
(201,632)(202,637)(203,639)(204,638)(205,634)(206,636)(207,635)(208,640)
(209,642)(210,641)(211,646)(212,648)(213,647)(214,643)(215,645)(216,644)
(217,595)(218,597)(219,596)(220,601)(221,603)(222,602)(223,598)(224,600)
(225,599)(226,604)(227,606)(228,605)(229,610)(230,612)(231,611)(232,607)
(233,609)(234,608)(235,613)(236,615)(237,614)(238,619)(239,621)(240,620)
(241,616)(242,618)(243,617)(244,487)(245,489)(246,488)(247,493)(248,495)
(249,494)(250,490)(251,492)(252,491)(253,496)(254,498)(255,497)(256,502)
(257,504)(258,503)(259,499)(260,501)(261,500)(262,505)(263,507)(264,506)
(265,511)(266,513)(267,512)(268,508)(269,510)(270,509)(271,541)(272,543)
(273,542)(274,547)(275,549)(276,548)(277,544)(278,546)(279,545)(280,550)
(281,552)(282,551)(283,556)(284,558)(285,557)(286,553)(287,555)(288,554)
(289,559)(290,561)(291,560)(292,565)(293,567)(294,566)(295,562)(296,564)
(297,563)(298,514)(299,516)(300,515)(301,520)(302,522)(303,521)(304,517)
(305,519)(306,518)(307,523)(308,525)(309,524)(310,529)(311,531)(312,530)
(313,526)(314,528)(315,527)(316,532)(317,534)(318,533)(319,538)(320,540)
(321,539)(322,535)(323,537)(324,536);;
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, s0*s1*s0*s1*s0*s1,
s1*s2*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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)( 2,488)( 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)!( 1,551)( 2,552)( 3,550)( 4,554)( 5,555)( 6,553)( 7,557)
( 8,558)( 9,556)( 10,543)( 11,541)( 12,542)( 13,546)( 14,544)( 15,545)
( 16,549)( 17,547)( 18,548)( 19,559)( 20,560)( 21,561)( 22,562)( 23,563)
( 24,564)( 25,565)( 26,566)( 27,567)( 28,524)( 29,525)( 30,523)( 31,527)
( 32,528)( 33,526)( 34,530)( 35,531)( 36,529)( 37,516)( 38,514)( 39,515)
( 40,519)( 41,517)( 42,518)( 43,522)( 44,520)( 45,521)( 46,532)( 47,533)
( 48,534)( 49,535)( 50,536)( 51,537)( 52,538)( 53,539)( 54,540)( 55,497)
( 56,498)( 57,496)( 58,500)( 59,501)( 60,499)( 61,503)( 62,504)( 63,502)
( 64,489)( 65,487)( 66,488)( 67,492)( 68,490)( 69,491)( 70,495)( 71,493)
( 72,494)( 73,505)( 74,506)( 75,507)( 76,508)( 77,509)( 78,510)( 79,511)
( 80,512)( 81,513)( 82,632)( 83,633)( 84,631)( 85,635)( 86,636)( 87,634)
( 88,638)( 89,639)( 90,637)( 91,624)( 92,622)( 93,623)( 94,627)( 95,625)
( 96,626)( 97,630)( 98,628)( 99,629)(100,640)(101,641)(102,642)(103,643)
(104,644)(105,645)(106,646)(107,647)(108,648)(109,605)(110,606)(111,604)
(112,608)(113,609)(114,607)(115,611)(116,612)(117,610)(118,597)(119,595)
(120,596)(121,600)(122,598)(123,599)(124,603)(125,601)(126,602)(127,613)
(128,614)(129,615)(130,616)(131,617)(132,618)(133,619)(134,620)(135,621)
(136,578)(137,579)(138,577)(139,581)(140,582)(141,580)(142,584)(143,585)
(144,583)(145,570)(146,568)(147,569)(148,573)(149,571)(150,572)(151,576)
(152,574)(153,575)(154,586)(155,587)(156,588)(157,589)(158,590)(159,591)
(160,592)(161,593)(162,594)(163,470)(164,471)(165,469)(166,473)(167,474)
(168,472)(169,476)(170,477)(171,475)(172,462)(173,460)(174,461)(175,465)
(176,463)(177,464)(178,468)(179,466)(180,467)(181,478)(182,479)(183,480)
(184,481)(185,482)(186,483)(187,484)(188,485)(189,486)(190,443)(191,444)
(192,442)(193,446)(194,447)(195,445)(196,449)(197,450)(198,448)(199,435)
(200,433)(201,434)(202,438)(203,436)(204,437)(205,441)(206,439)(207,440)
(208,451)(209,452)(210,453)(211,454)(212,455)(213,456)(214,457)(215,458)
(216,459)(217,416)(218,417)(219,415)(220,419)(221,420)(222,418)(223,422)
(224,423)(225,421)(226,408)(227,406)(228,407)(229,411)(230,409)(231,410)
(232,414)(233,412)(234,413)(235,424)(236,425)(237,426)(238,427)(239,428)
(240,429)(241,430)(242,431)(243,432)(244,389)(245,390)(246,388)(247,392)
(248,393)(249,391)(250,395)(251,396)(252,394)(253,381)(254,379)(255,380)
(256,384)(257,382)(258,383)(259,387)(260,385)(261,386)(262,397)(263,398)
(264,399)(265,400)(266,401)(267,402)(268,403)(269,404)(270,405)(271,362)
(272,363)(273,361)(274,365)(275,366)(276,364)(277,368)(278,369)(279,367)
(280,354)(281,352)(282,353)(283,357)(284,355)(285,356)(286,360)(287,358)
(288,359)(289,370)(290,371)(291,372)(292,373)(293,374)(294,375)(295,376)
(296,377)(297,378)(298,335)(299,336)(300,334)(301,338)(302,339)(303,337)
(304,341)(305,342)(306,340)(307,327)(308,325)(309,326)(310,330)(311,328)
(312,329)(313,333)(314,331)(315,332)(316,343)(317,344)(318,345)(319,346)
(320,347)(321,348)(322,349)(323,350)(324,351);
s2 := Sym(648)!( 1,325)( 2,327)( 3,326)( 4,331)( 5,333)( 6,332)( 7,328)
( 8,330)( 9,329)( 10,334)( 11,336)( 12,335)( 13,340)( 14,342)( 15,341)
( 16,337)( 17,339)( 18,338)( 19,343)( 20,345)( 21,344)( 22,349)( 23,351)
( 24,350)( 25,346)( 26,348)( 27,347)( 28,379)( 29,381)( 30,380)( 31,385)
( 32,387)( 33,386)( 34,382)( 35,384)( 36,383)( 37,388)( 38,390)( 39,389)
( 40,394)( 41,396)( 42,395)( 43,391)( 44,393)( 45,392)( 46,397)( 47,399)
( 48,398)( 49,403)( 50,405)( 51,404)( 52,400)( 53,402)( 54,401)( 55,352)
( 56,354)( 57,353)( 58,358)( 59,360)( 60,359)( 61,355)( 62,357)( 63,356)
( 64,361)( 65,363)( 66,362)( 67,367)( 68,369)( 69,368)( 70,364)( 71,366)
( 72,365)( 73,370)( 74,372)( 75,371)( 76,376)( 77,378)( 78,377)( 79,373)
( 80,375)( 81,374)( 82,406)( 83,408)( 84,407)( 85,412)( 86,414)( 87,413)
( 88,409)( 89,411)( 90,410)( 91,415)( 92,417)( 93,416)( 94,421)( 95,423)
( 96,422)( 97,418)( 98,420)( 99,419)(100,424)(101,426)(102,425)(103,430)
(104,432)(105,431)(106,427)(107,429)(108,428)(109,460)(110,462)(111,461)
(112,466)(113,468)(114,467)(115,463)(116,465)(117,464)(118,469)(119,471)
(120,470)(121,475)(122,477)(123,476)(124,472)(125,474)(126,473)(127,478)
(128,480)(129,479)(130,484)(131,486)(132,485)(133,481)(134,483)(135,482)
(136,433)(137,435)(138,434)(139,439)(140,441)(141,440)(142,436)(143,438)
(144,437)(145,442)(146,444)(147,443)(148,448)(149,450)(150,449)(151,445)
(152,447)(153,446)(154,451)(155,453)(156,452)(157,457)(158,459)(159,458)
(160,454)(161,456)(162,455)(163,568)(164,570)(165,569)(166,574)(167,576)
(168,575)(169,571)(170,573)(171,572)(172,577)(173,579)(174,578)(175,583)
(176,585)(177,584)(178,580)(179,582)(180,581)(181,586)(182,588)(183,587)
(184,592)(185,594)(186,593)(187,589)(188,591)(189,590)(190,622)(191,624)
(192,623)(193,628)(194,630)(195,629)(196,625)(197,627)(198,626)(199,631)
(200,633)(201,632)(202,637)(203,639)(204,638)(205,634)(206,636)(207,635)
(208,640)(209,642)(210,641)(211,646)(212,648)(213,647)(214,643)(215,645)
(216,644)(217,595)(218,597)(219,596)(220,601)(221,603)(222,602)(223,598)
(224,600)(225,599)(226,604)(227,606)(228,605)(229,610)(230,612)(231,611)
(232,607)(233,609)(234,608)(235,613)(236,615)(237,614)(238,619)(239,621)
(240,620)(241,616)(242,618)(243,617)(244,487)(245,489)(246,488)(247,493)
(248,495)(249,494)(250,490)(251,492)(252,491)(253,496)(254,498)(255,497)
(256,502)(257,504)(258,503)(259,499)(260,501)(261,500)(262,505)(263,507)
(264,506)(265,511)(266,513)(267,512)(268,508)(269,510)(270,509)(271,541)
(272,543)(273,542)(274,547)(275,549)(276,548)(277,544)(278,546)(279,545)
(280,550)(281,552)(282,551)(283,556)(284,558)(285,557)(286,553)(287,555)
(288,554)(289,559)(290,561)(291,560)(292,565)(293,567)(294,566)(295,562)
(296,564)(297,563)(298,514)(299,516)(300,515)(301,520)(302,522)(303,521)
(304,517)(305,519)(306,518)(307,523)(308,525)(309,524)(310,529)(311,531)
(312,530)(313,526)(314,528)(315,527)(316,532)(317,534)(318,533)(319,538)
(320,540)(321,539)(322,535)(323,537)(324,536);
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,
s0*s1*s0*s1*s0*s1, s1*s2*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*s0*s1*s2*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