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.
Irregular Quotients (of which this is a minimal cover):
P/N, where N=<s2*s3*s2*s3*s2*s3*s2*s3> of order 3.
12 facets:
12 of {3,6}*36
3 vertex figures:
3 of 3-fold non-regular quotient of {6,12}*432b
P/N, where N=<s2*s1*s3*s2*s1*s2*s3*s2> of order 3.
20 facets:
8 of {3,6}*36
12 of {3,2}*12
3 vertex figures:
3 of 3-fold non-regular quotient of {6,12}*432b
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