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# Polytope of Type {12,6,3}

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

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

```
References : None.
to this polytope