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Polytope of Type {12,6,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,12}*1728c
if this polytope has a name.
Group : SmallGroup(1728,36991)
Rank : 4
Schlafli Type : {12,6,12}
Number of vertices, edges, etc : 12, 36, 36, 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 : {6,6,12}*864d, {12,6,6}*864f
   3-fold quotients : {4,6,12}*576a, {12,6,4}*576c
   4-fold quotients : {6,6,6}*432d
   6-fold quotients : {2,6,12}*288a, {4,6,6}*288a, {6,6,4}*288b, {12,6,2}*288c
   9-fold quotients : {4,2,12}*192, {4,6,4}*192a
   12-fold quotients : {2,6,6}*144a, {6,6,2}*144b
   18-fold quotients : {2,2,12}*96, {2,6,4}*96a, {4,2,6}*96, {4,6,2}*96a
   24-fold quotients : {6,3,2}*72
   27-fold quotients : {4,2,4}*64
   36-fold quotients : {4,2,3}*48, {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,2,4}*32, {4,2,2}*32
   72-fold quotients : {2,2,3}*24, {2,3,2}*24
   108-fold quotients : {2,2,2}*16
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)(109,136)(110,137)(111,138)(112,142)
(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)(120,147)
(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)(128,155)
(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)(163,190)
(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)(171,195)
(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)(179,203)
(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)(187,211)
(188,212)(189,213)(220,223)(221,224)(222,225)(229,232)(230,233)(231,234)
(238,241)(239,242)(240,243)(247,250)(248,251)(249,252)(256,259)(257,260)
(258,261)(265,268)(266,269)(267,270)(274,277)(275,278)(276,279)(283,286)
(284,287)(285,288)(292,295)(293,296)(294,297)(301,304)(302,305)(303,306)
(310,313)(311,314)(312,315)(319,322)(320,323)(321,324)(325,352)(326,353)
(327,354)(328,358)(329,359)(330,360)(331,355)(332,356)(333,357)(334,361)
(335,362)(336,363)(337,367)(338,368)(339,369)(340,364)(341,365)(342,366)
(343,370)(344,371)(345,372)(346,376)(347,377)(348,378)(349,373)(350,374)
(351,375)(379,406)(380,407)(381,408)(382,412)(383,413)(384,414)(385,409)
(386,410)(387,411)(388,415)(389,416)(390,417)(391,421)(392,422)(393,423)
(394,418)(395,419)(396,420)(397,424)(398,425)(399,426)(400,430)(401,431)
(402,432)(403,427)(404,428)(405,429)(433,460)(434,461)(435,462)(436,466)
(437,467)(438,468)(439,463)(440,464)(441,465)(442,469)(443,470)(444,471)
(445,475)(446,476)(447,477)(448,472)(449,473)(450,474)(451,478)(452,479)
(453,480)(454,484)(455,485)(456,486)(457,481)(458,482)(459,483)(487,514)
(488,515)(489,516)(490,520)(491,521)(492,522)(493,517)(494,518)(495,519)
(496,523)(497,524)(498,525)(499,529)(500,530)(501,531)(502,526)(503,527)
(504,528)(505,532)(506,533)(507,534)(508,538)(509,539)(510,540)(511,535)
(512,536)(513,537)(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)(649,676)(650,677)
(651,678)(652,682)(653,683)(654,684)(655,679)(656,680)(657,681)(658,685)
(659,686)(660,687)(661,691)(662,692)(663,693)(664,688)(665,689)(666,690)
(667,694)(668,695)(669,696)(670,700)(671,701)(672,702)(673,697)(674,698)
(675,699)(703,730)(704,731)(705,732)(706,736)(707,737)(708,738)(709,733)
(710,734)(711,735)(712,739)(713,740)(714,741)(715,745)(716,746)(717,747)
(718,742)(719,743)(720,744)(721,748)(722,749)(723,750)(724,754)(725,755)
(726,756)(727,751)(728,752)(729,753)(760,763)(761,764)(762,765)(769,772)
(770,773)(771,774)(778,781)(779,782)(780,783)(787,790)(788,791)(789,792)
(796,799)(797,800)(798,801)(805,808)(806,809)(807,810)(814,817)(815,818)
(816,819)(823,826)(824,827)(825,828)(832,835)(833,836)(834,837)(841,844)
(842,845)(843,846)(850,853)(851,854)(852,855)(859,862)(860,863)(861,864);;
s1 := (  1,436)(  2,437)(  3,438)(  4,433)(  5,434)(  6,435)(  7,439)(  8,440)
(  9,441)( 10,454)( 11,455)( 12,456)( 13,451)( 14,452)( 15,453)( 16,457)
( 17,458)( 18,459)( 19,445)( 20,446)( 21,447)( 22,442)( 23,443)( 24,444)
( 25,448)( 26,449)( 27,450)( 28,463)( 29,464)( 30,465)( 31,460)( 32,461)
( 33,462)( 34,466)( 35,467)( 36,468)( 37,481)( 38,482)( 39,483)( 40,478)
( 41,479)( 42,480)( 43,484)( 44,485)( 45,486)( 46,472)( 47,473)( 48,474)
( 49,469)( 50,470)( 51,471)( 52,475)( 53,476)( 54,477)( 55,490)( 56,491)
( 57,492)( 58,487)( 59,488)( 60,489)( 61,493)( 62,494)( 63,495)( 64,508)
( 65,509)( 66,510)( 67,505)( 68,506)( 69,507)( 70,511)( 71,512)( 72,513)
( 73,499)( 74,500)( 75,501)( 76,496)( 77,497)( 78,498)( 79,502)( 80,503)
( 81,504)( 82,517)( 83,518)( 84,519)( 85,514)( 86,515)( 87,516)( 88,520)
( 89,521)( 90,522)( 91,535)( 92,536)( 93,537)( 94,532)( 95,533)( 96,534)
( 97,538)( 98,539)( 99,540)(100,526)(101,527)(102,528)(103,523)(104,524)
(105,525)(106,529)(107,530)(108,531)(109,544)(110,545)(111,546)(112,541)
(113,542)(114,543)(115,547)(116,548)(117,549)(118,562)(119,563)(120,564)
(121,559)(122,560)(123,561)(124,565)(125,566)(126,567)(127,553)(128,554)
(129,555)(130,550)(131,551)(132,552)(133,556)(134,557)(135,558)(136,571)
(137,572)(138,573)(139,568)(140,569)(141,570)(142,574)(143,575)(144,576)
(145,589)(146,590)(147,591)(148,586)(149,587)(150,588)(151,592)(152,593)
(153,594)(154,580)(155,581)(156,582)(157,577)(158,578)(159,579)(160,583)
(161,584)(162,585)(163,598)(164,599)(165,600)(166,595)(167,596)(168,597)
(169,601)(170,602)(171,603)(172,616)(173,617)(174,618)(175,613)(176,614)
(177,615)(178,619)(179,620)(180,621)(181,607)(182,608)(183,609)(184,604)
(185,605)(186,606)(187,610)(188,611)(189,612)(190,625)(191,626)(192,627)
(193,622)(194,623)(195,624)(196,628)(197,629)(198,630)(199,643)(200,644)
(201,645)(202,640)(203,641)(204,642)(205,646)(206,647)(207,648)(208,634)
(209,635)(210,636)(211,631)(212,632)(213,633)(214,637)(215,638)(216,639)
(217,652)(218,653)(219,654)(220,649)(221,650)(222,651)(223,655)(224,656)
(225,657)(226,670)(227,671)(228,672)(229,667)(230,668)(231,669)(232,673)
(233,674)(234,675)(235,661)(236,662)(237,663)(238,658)(239,659)(240,660)
(241,664)(242,665)(243,666)(244,679)(245,680)(246,681)(247,676)(248,677)
(249,678)(250,682)(251,683)(252,684)(253,697)(254,698)(255,699)(256,694)
(257,695)(258,696)(259,700)(260,701)(261,702)(262,688)(263,689)(264,690)
(265,685)(266,686)(267,687)(268,691)(269,692)(270,693)(271,706)(272,707)
(273,708)(274,703)(275,704)(276,705)(277,709)(278,710)(279,711)(280,724)
(281,725)(282,726)(283,721)(284,722)(285,723)(286,727)(287,728)(288,729)
(289,715)(290,716)(291,717)(292,712)(293,713)(294,714)(295,718)(296,719)
(297,720)(298,733)(299,734)(300,735)(301,730)(302,731)(303,732)(304,736)
(305,737)(306,738)(307,751)(308,752)(309,753)(310,748)(311,749)(312,750)
(313,754)(314,755)(315,756)(316,742)(317,743)(318,744)(319,739)(320,740)
(321,741)(322,745)(323,746)(324,747)(325,760)(326,761)(327,762)(328,757)
(329,758)(330,759)(331,763)(332,764)(333,765)(334,778)(335,779)(336,780)
(337,775)(338,776)(339,777)(340,781)(341,782)(342,783)(343,769)(344,770)
(345,771)(346,766)(347,767)(348,768)(349,772)(350,773)(351,774)(352,787)
(353,788)(354,789)(355,784)(356,785)(357,786)(358,790)(359,791)(360,792)
(361,805)(362,806)(363,807)(364,802)(365,803)(366,804)(367,808)(368,809)
(369,810)(370,796)(371,797)(372,798)(373,793)(374,794)(375,795)(376,799)
(377,800)(378,801)(379,814)(380,815)(381,816)(382,811)(383,812)(384,813)
(385,817)(386,818)(387,819)(388,832)(389,833)(390,834)(391,829)(392,830)
(393,831)(394,835)(395,836)(396,837)(397,823)(398,824)(399,825)(400,820)
(401,821)(402,822)(403,826)(404,827)(405,828)(406,841)(407,842)(408,843)
(409,838)(410,839)(411,840)(412,844)(413,845)(414,846)(415,859)(416,860)
(417,861)(418,856)(419,857)(420,858)(421,862)(422,863)(423,864)(424,850)
(425,851)(426,852)(427,847)(428,848)(429,849)(430,853)(431,854)(432,855);;
s2 := (  1,766)(  2,768)(  3,767)(  4,772)(  5,774)(  6,773)(  7,769)(  8,771)
(  9,770)( 10,757)( 11,759)( 12,758)( 13,763)( 14,765)( 15,764)( 16,760)
( 17,762)( 18,761)( 19,775)( 20,777)( 21,776)( 22,781)( 23,783)( 24,782)
( 25,778)( 26,780)( 27,779)( 28,793)( 29,795)( 30,794)( 31,799)( 32,801)
( 33,800)( 34,796)( 35,798)( 36,797)( 37,784)( 38,786)( 39,785)( 40,790)
( 41,792)( 42,791)( 43,787)( 44,789)( 45,788)( 46,802)( 47,804)( 48,803)
( 49,808)( 50,810)( 51,809)( 52,805)( 53,807)( 54,806)( 55,820)( 56,822)
( 57,821)( 58,826)( 59,828)( 60,827)( 61,823)( 62,825)( 63,824)( 64,811)
( 65,813)( 66,812)( 67,817)( 68,819)( 69,818)( 70,814)( 71,816)( 72,815)
( 73,829)( 74,831)( 75,830)( 76,835)( 77,837)( 78,836)( 79,832)( 80,834)
( 81,833)( 82,847)( 83,849)( 84,848)( 85,853)( 86,855)( 87,854)( 88,850)
( 89,852)( 90,851)( 91,838)( 92,840)( 93,839)( 94,844)( 95,846)( 96,845)
( 97,841)( 98,843)( 99,842)(100,856)(101,858)(102,857)(103,862)(104,864)
(105,863)(106,859)(107,861)(108,860)(109,658)(110,660)(111,659)(112,664)
(113,666)(114,665)(115,661)(116,663)(117,662)(118,649)(119,651)(120,650)
(121,655)(122,657)(123,656)(124,652)(125,654)(126,653)(127,667)(128,669)
(129,668)(130,673)(131,675)(132,674)(133,670)(134,672)(135,671)(136,685)
(137,687)(138,686)(139,691)(140,693)(141,692)(142,688)(143,690)(144,689)
(145,676)(146,678)(147,677)(148,682)(149,684)(150,683)(151,679)(152,681)
(153,680)(154,694)(155,696)(156,695)(157,700)(158,702)(159,701)(160,697)
(161,699)(162,698)(163,712)(164,714)(165,713)(166,718)(167,720)(168,719)
(169,715)(170,717)(171,716)(172,703)(173,705)(174,704)(175,709)(176,711)
(177,710)(178,706)(179,708)(180,707)(181,721)(182,723)(183,722)(184,727)
(185,729)(186,728)(187,724)(188,726)(189,725)(190,739)(191,741)(192,740)
(193,745)(194,747)(195,746)(196,742)(197,744)(198,743)(199,730)(200,732)
(201,731)(202,736)(203,738)(204,737)(205,733)(206,735)(207,734)(208,748)
(209,750)(210,749)(211,754)(212,756)(213,755)(214,751)(215,753)(216,752)
(217,550)(218,552)(219,551)(220,556)(221,558)(222,557)(223,553)(224,555)
(225,554)(226,541)(227,543)(228,542)(229,547)(230,549)(231,548)(232,544)
(233,546)(234,545)(235,559)(236,561)(237,560)(238,565)(239,567)(240,566)
(241,562)(242,564)(243,563)(244,577)(245,579)(246,578)(247,583)(248,585)
(249,584)(250,580)(251,582)(252,581)(253,568)(254,570)(255,569)(256,574)
(257,576)(258,575)(259,571)(260,573)(261,572)(262,586)(263,588)(264,587)
(265,592)(266,594)(267,593)(268,589)(269,591)(270,590)(271,604)(272,606)
(273,605)(274,610)(275,612)(276,611)(277,607)(278,609)(279,608)(280,595)
(281,597)(282,596)(283,601)(284,603)(285,602)(286,598)(287,600)(288,599)
(289,613)(290,615)(291,614)(292,619)(293,621)(294,620)(295,616)(296,618)
(297,617)(298,631)(299,633)(300,632)(301,637)(302,639)(303,638)(304,634)
(305,636)(306,635)(307,622)(308,624)(309,623)(310,628)(311,630)(312,629)
(313,625)(314,627)(315,626)(316,640)(317,642)(318,641)(319,646)(320,648)
(321,647)(322,643)(323,645)(324,644)(325,442)(326,444)(327,443)(328,448)
(329,450)(330,449)(331,445)(332,447)(333,446)(334,433)(335,435)(336,434)
(337,439)(338,441)(339,440)(340,436)(341,438)(342,437)(343,451)(344,453)
(345,452)(346,457)(347,459)(348,458)(349,454)(350,456)(351,455)(352,469)
(353,471)(354,470)(355,475)(356,477)(357,476)(358,472)(359,474)(360,473)
(361,460)(362,462)(363,461)(364,466)(365,468)(366,467)(367,463)(368,465)
(369,464)(370,478)(371,480)(372,479)(373,484)(374,486)(375,485)(376,481)
(377,483)(378,482)(379,496)(380,498)(381,497)(382,502)(383,504)(384,503)
(385,499)(386,501)(387,500)(388,487)(389,489)(390,488)(391,493)(392,495)
(393,494)(394,490)(395,492)(396,491)(397,505)(398,507)(399,506)(400,511)
(401,513)(402,512)(403,508)(404,510)(405,509)(406,523)(407,525)(408,524)
(409,529)(410,531)(411,530)(412,526)(413,528)(414,527)(415,514)(416,516)
(417,515)(418,520)(419,522)(420,521)(421,517)(422,519)(423,518)(424,532)
(425,534)(426,533)(427,538)(428,540)(429,539)(430,535)(431,537)(432,536);;
s3 := (  1,218)(  2,217)(  3,219)(  4,221)(  5,220)(  6,222)(  7,224)(  8,223)
(  9,225)( 10,227)( 11,226)( 12,228)( 13,230)( 14,229)( 15,231)( 16,233)
( 17,232)( 18,234)( 19,236)( 20,235)( 21,237)( 22,239)( 23,238)( 24,240)
( 25,242)( 26,241)( 27,243)( 28,245)( 29,244)( 30,246)( 31,248)( 32,247)
( 33,249)( 34,251)( 35,250)( 36,252)( 37,254)( 38,253)( 39,255)( 40,257)
( 41,256)( 42,258)( 43,260)( 44,259)( 45,261)( 46,263)( 47,262)( 48,264)
( 49,266)( 50,265)( 51,267)( 52,269)( 53,268)( 54,270)( 55,272)( 56,271)
( 57,273)( 58,275)( 59,274)( 60,276)( 61,278)( 62,277)( 63,279)( 64,281)
( 65,280)( 66,282)( 67,284)( 68,283)( 69,285)( 70,287)( 71,286)( 72,288)
( 73,290)( 74,289)( 75,291)( 76,293)( 77,292)( 78,294)( 79,296)( 80,295)
( 81,297)( 82,299)( 83,298)( 84,300)( 85,302)( 86,301)( 87,303)( 88,305)
( 89,304)( 90,306)( 91,308)( 92,307)( 93,309)( 94,311)( 95,310)( 96,312)
( 97,314)( 98,313)( 99,315)(100,317)(101,316)(102,318)(103,320)(104,319)
(105,321)(106,323)(107,322)(108,324)(109,380)(110,379)(111,381)(112,383)
(113,382)(114,384)(115,386)(116,385)(117,387)(118,389)(119,388)(120,390)
(121,392)(122,391)(123,393)(124,395)(125,394)(126,396)(127,398)(128,397)
(129,399)(130,401)(131,400)(132,402)(133,404)(134,403)(135,405)(136,407)
(137,406)(138,408)(139,410)(140,409)(141,411)(142,413)(143,412)(144,414)
(145,416)(146,415)(147,417)(148,419)(149,418)(150,420)(151,422)(152,421)
(153,423)(154,425)(155,424)(156,426)(157,428)(158,427)(159,429)(160,431)
(161,430)(162,432)(163,326)(164,325)(165,327)(166,329)(167,328)(168,330)
(169,332)(170,331)(171,333)(172,335)(173,334)(174,336)(175,338)(176,337)
(177,339)(178,341)(179,340)(180,342)(181,344)(182,343)(183,345)(184,347)
(185,346)(186,348)(187,350)(188,349)(189,351)(190,353)(191,352)(192,354)
(193,356)(194,355)(195,357)(196,359)(197,358)(198,360)(199,362)(200,361)
(201,363)(202,365)(203,364)(204,366)(205,368)(206,367)(207,369)(208,371)
(209,370)(210,372)(211,374)(212,373)(213,375)(214,377)(215,376)(216,378)
(433,650)(434,649)(435,651)(436,653)(437,652)(438,654)(439,656)(440,655)
(441,657)(442,659)(443,658)(444,660)(445,662)(446,661)(447,663)(448,665)
(449,664)(450,666)(451,668)(452,667)(453,669)(454,671)(455,670)(456,672)
(457,674)(458,673)(459,675)(460,677)(461,676)(462,678)(463,680)(464,679)
(465,681)(466,683)(467,682)(468,684)(469,686)(470,685)(471,687)(472,689)
(473,688)(474,690)(475,692)(476,691)(477,693)(478,695)(479,694)(480,696)
(481,698)(482,697)(483,699)(484,701)(485,700)(486,702)(487,704)(488,703)
(489,705)(490,707)(491,706)(492,708)(493,710)(494,709)(495,711)(496,713)
(497,712)(498,714)(499,716)(500,715)(501,717)(502,719)(503,718)(504,720)
(505,722)(506,721)(507,723)(508,725)(509,724)(510,726)(511,728)(512,727)
(513,729)(514,731)(515,730)(516,732)(517,734)(518,733)(519,735)(520,737)
(521,736)(522,738)(523,740)(524,739)(525,741)(526,743)(527,742)(528,744)
(529,746)(530,745)(531,747)(532,749)(533,748)(534,750)(535,752)(536,751)
(537,753)(538,755)(539,754)(540,756)(541,812)(542,811)(543,813)(544,815)
(545,814)(546,816)(547,818)(548,817)(549,819)(550,821)(551,820)(552,822)
(553,824)(554,823)(555,825)(556,827)(557,826)(558,828)(559,830)(560,829)
(561,831)(562,833)(563,832)(564,834)(565,836)(566,835)(567,837)(568,839)
(569,838)(570,840)(571,842)(572,841)(573,843)(574,845)(575,844)(576,846)
(577,848)(578,847)(579,849)(580,851)(581,850)(582,852)(583,854)(584,853)
(585,855)(586,857)(587,856)(588,858)(589,860)(590,859)(591,861)(592,863)
(593,862)(594,864)(595,758)(596,757)(597,759)(598,761)(599,760)(600,762)
(601,764)(602,763)(603,765)(604,767)(605,766)(606,768)(607,770)(608,769)
(609,771)(610,773)(611,772)(612,774)(613,776)(614,775)(615,777)(616,779)
(617,778)(618,780)(619,782)(620,781)(621,783)(622,785)(623,784)(624,786)
(625,788)(626,787)(627,789)(628,791)(629,790)(630,792)(631,794)(632,793)
(633,795)(634,797)(635,796)(636,798)(637,800)(638,799)(639,801)(640,803)
(641,802)(642,804)(643,806)(644,805)(645,807)(646,809)(647,808)(648,810);;
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, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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(864)!(  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)(109,136)(110,137)(111,138)
(112,142)(113,143)(114,144)(115,139)(116,140)(117,141)(118,145)(119,146)
(120,147)(121,151)(122,152)(123,153)(124,148)(125,149)(126,150)(127,154)
(128,155)(129,156)(130,160)(131,161)(132,162)(133,157)(134,158)(135,159)
(163,190)(164,191)(165,192)(166,196)(167,197)(168,198)(169,193)(170,194)
(171,195)(172,199)(173,200)(174,201)(175,205)(176,206)(177,207)(178,202)
(179,203)(180,204)(181,208)(182,209)(183,210)(184,214)(185,215)(186,216)
(187,211)(188,212)(189,213)(220,223)(221,224)(222,225)(229,232)(230,233)
(231,234)(238,241)(239,242)(240,243)(247,250)(248,251)(249,252)(256,259)
(257,260)(258,261)(265,268)(266,269)(267,270)(274,277)(275,278)(276,279)
(283,286)(284,287)(285,288)(292,295)(293,296)(294,297)(301,304)(302,305)
(303,306)(310,313)(311,314)(312,315)(319,322)(320,323)(321,324)(325,352)
(326,353)(327,354)(328,358)(329,359)(330,360)(331,355)(332,356)(333,357)
(334,361)(335,362)(336,363)(337,367)(338,368)(339,369)(340,364)(341,365)
(342,366)(343,370)(344,371)(345,372)(346,376)(347,377)(348,378)(349,373)
(350,374)(351,375)(379,406)(380,407)(381,408)(382,412)(383,413)(384,414)
(385,409)(386,410)(387,411)(388,415)(389,416)(390,417)(391,421)(392,422)
(393,423)(394,418)(395,419)(396,420)(397,424)(398,425)(399,426)(400,430)
(401,431)(402,432)(403,427)(404,428)(405,429)(433,460)(434,461)(435,462)
(436,466)(437,467)(438,468)(439,463)(440,464)(441,465)(442,469)(443,470)
(444,471)(445,475)(446,476)(447,477)(448,472)(449,473)(450,474)(451,478)
(452,479)(453,480)(454,484)(455,485)(456,486)(457,481)(458,482)(459,483)
(487,514)(488,515)(489,516)(490,520)(491,521)(492,522)(493,517)(494,518)
(495,519)(496,523)(497,524)(498,525)(499,529)(500,530)(501,531)(502,526)
(503,527)(504,528)(505,532)(506,533)(507,534)(508,538)(509,539)(510,540)
(511,535)(512,536)(513,537)(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)(649,676)
(650,677)(651,678)(652,682)(653,683)(654,684)(655,679)(656,680)(657,681)
(658,685)(659,686)(660,687)(661,691)(662,692)(663,693)(664,688)(665,689)
(666,690)(667,694)(668,695)(669,696)(670,700)(671,701)(672,702)(673,697)
(674,698)(675,699)(703,730)(704,731)(705,732)(706,736)(707,737)(708,738)
(709,733)(710,734)(711,735)(712,739)(713,740)(714,741)(715,745)(716,746)
(717,747)(718,742)(719,743)(720,744)(721,748)(722,749)(723,750)(724,754)
(725,755)(726,756)(727,751)(728,752)(729,753)(760,763)(761,764)(762,765)
(769,772)(770,773)(771,774)(778,781)(779,782)(780,783)(787,790)(788,791)
(789,792)(796,799)(797,800)(798,801)(805,808)(806,809)(807,810)(814,817)
(815,818)(816,819)(823,826)(824,827)(825,828)(832,835)(833,836)(834,837)
(841,844)(842,845)(843,846)(850,853)(851,854)(852,855)(859,862)(860,863)
(861,864);
s1 := Sym(864)!(  1,436)(  2,437)(  3,438)(  4,433)(  5,434)(  6,435)(  7,439)
(  8,440)(  9,441)( 10,454)( 11,455)( 12,456)( 13,451)( 14,452)( 15,453)
( 16,457)( 17,458)( 18,459)( 19,445)( 20,446)( 21,447)( 22,442)( 23,443)
( 24,444)( 25,448)( 26,449)( 27,450)( 28,463)( 29,464)( 30,465)( 31,460)
( 32,461)( 33,462)( 34,466)( 35,467)( 36,468)( 37,481)( 38,482)( 39,483)
( 40,478)( 41,479)( 42,480)( 43,484)( 44,485)( 45,486)( 46,472)( 47,473)
( 48,474)( 49,469)( 50,470)( 51,471)( 52,475)( 53,476)( 54,477)( 55,490)
( 56,491)( 57,492)( 58,487)( 59,488)( 60,489)( 61,493)( 62,494)( 63,495)
( 64,508)( 65,509)( 66,510)( 67,505)( 68,506)( 69,507)( 70,511)( 71,512)
( 72,513)( 73,499)( 74,500)( 75,501)( 76,496)( 77,497)( 78,498)( 79,502)
( 80,503)( 81,504)( 82,517)( 83,518)( 84,519)( 85,514)( 86,515)( 87,516)
( 88,520)( 89,521)( 90,522)( 91,535)( 92,536)( 93,537)( 94,532)( 95,533)
( 96,534)( 97,538)( 98,539)( 99,540)(100,526)(101,527)(102,528)(103,523)
(104,524)(105,525)(106,529)(107,530)(108,531)(109,544)(110,545)(111,546)
(112,541)(113,542)(114,543)(115,547)(116,548)(117,549)(118,562)(119,563)
(120,564)(121,559)(122,560)(123,561)(124,565)(125,566)(126,567)(127,553)
(128,554)(129,555)(130,550)(131,551)(132,552)(133,556)(134,557)(135,558)
(136,571)(137,572)(138,573)(139,568)(140,569)(141,570)(142,574)(143,575)
(144,576)(145,589)(146,590)(147,591)(148,586)(149,587)(150,588)(151,592)
(152,593)(153,594)(154,580)(155,581)(156,582)(157,577)(158,578)(159,579)
(160,583)(161,584)(162,585)(163,598)(164,599)(165,600)(166,595)(167,596)
(168,597)(169,601)(170,602)(171,603)(172,616)(173,617)(174,618)(175,613)
(176,614)(177,615)(178,619)(179,620)(180,621)(181,607)(182,608)(183,609)
(184,604)(185,605)(186,606)(187,610)(188,611)(189,612)(190,625)(191,626)
(192,627)(193,622)(194,623)(195,624)(196,628)(197,629)(198,630)(199,643)
(200,644)(201,645)(202,640)(203,641)(204,642)(205,646)(206,647)(207,648)
(208,634)(209,635)(210,636)(211,631)(212,632)(213,633)(214,637)(215,638)
(216,639)(217,652)(218,653)(219,654)(220,649)(221,650)(222,651)(223,655)
(224,656)(225,657)(226,670)(227,671)(228,672)(229,667)(230,668)(231,669)
(232,673)(233,674)(234,675)(235,661)(236,662)(237,663)(238,658)(239,659)
(240,660)(241,664)(242,665)(243,666)(244,679)(245,680)(246,681)(247,676)
(248,677)(249,678)(250,682)(251,683)(252,684)(253,697)(254,698)(255,699)
(256,694)(257,695)(258,696)(259,700)(260,701)(261,702)(262,688)(263,689)
(264,690)(265,685)(266,686)(267,687)(268,691)(269,692)(270,693)(271,706)
(272,707)(273,708)(274,703)(275,704)(276,705)(277,709)(278,710)(279,711)
(280,724)(281,725)(282,726)(283,721)(284,722)(285,723)(286,727)(287,728)
(288,729)(289,715)(290,716)(291,717)(292,712)(293,713)(294,714)(295,718)
(296,719)(297,720)(298,733)(299,734)(300,735)(301,730)(302,731)(303,732)
(304,736)(305,737)(306,738)(307,751)(308,752)(309,753)(310,748)(311,749)
(312,750)(313,754)(314,755)(315,756)(316,742)(317,743)(318,744)(319,739)
(320,740)(321,741)(322,745)(323,746)(324,747)(325,760)(326,761)(327,762)
(328,757)(329,758)(330,759)(331,763)(332,764)(333,765)(334,778)(335,779)
(336,780)(337,775)(338,776)(339,777)(340,781)(341,782)(342,783)(343,769)
(344,770)(345,771)(346,766)(347,767)(348,768)(349,772)(350,773)(351,774)
(352,787)(353,788)(354,789)(355,784)(356,785)(357,786)(358,790)(359,791)
(360,792)(361,805)(362,806)(363,807)(364,802)(365,803)(366,804)(367,808)
(368,809)(369,810)(370,796)(371,797)(372,798)(373,793)(374,794)(375,795)
(376,799)(377,800)(378,801)(379,814)(380,815)(381,816)(382,811)(383,812)
(384,813)(385,817)(386,818)(387,819)(388,832)(389,833)(390,834)(391,829)
(392,830)(393,831)(394,835)(395,836)(396,837)(397,823)(398,824)(399,825)
(400,820)(401,821)(402,822)(403,826)(404,827)(405,828)(406,841)(407,842)
(408,843)(409,838)(410,839)(411,840)(412,844)(413,845)(414,846)(415,859)
(416,860)(417,861)(418,856)(419,857)(420,858)(421,862)(422,863)(423,864)
(424,850)(425,851)(426,852)(427,847)(428,848)(429,849)(430,853)(431,854)
(432,855);
s2 := Sym(864)!(  1,766)(  2,768)(  3,767)(  4,772)(  5,774)(  6,773)(  7,769)
(  8,771)(  9,770)( 10,757)( 11,759)( 12,758)( 13,763)( 14,765)( 15,764)
( 16,760)( 17,762)( 18,761)( 19,775)( 20,777)( 21,776)( 22,781)( 23,783)
( 24,782)( 25,778)( 26,780)( 27,779)( 28,793)( 29,795)( 30,794)( 31,799)
( 32,801)( 33,800)( 34,796)( 35,798)( 36,797)( 37,784)( 38,786)( 39,785)
( 40,790)( 41,792)( 42,791)( 43,787)( 44,789)( 45,788)( 46,802)( 47,804)
( 48,803)( 49,808)( 50,810)( 51,809)( 52,805)( 53,807)( 54,806)( 55,820)
( 56,822)( 57,821)( 58,826)( 59,828)( 60,827)( 61,823)( 62,825)( 63,824)
( 64,811)( 65,813)( 66,812)( 67,817)( 68,819)( 69,818)( 70,814)( 71,816)
( 72,815)( 73,829)( 74,831)( 75,830)( 76,835)( 77,837)( 78,836)( 79,832)
( 80,834)( 81,833)( 82,847)( 83,849)( 84,848)( 85,853)( 86,855)( 87,854)
( 88,850)( 89,852)( 90,851)( 91,838)( 92,840)( 93,839)( 94,844)( 95,846)
( 96,845)( 97,841)( 98,843)( 99,842)(100,856)(101,858)(102,857)(103,862)
(104,864)(105,863)(106,859)(107,861)(108,860)(109,658)(110,660)(111,659)
(112,664)(113,666)(114,665)(115,661)(116,663)(117,662)(118,649)(119,651)
(120,650)(121,655)(122,657)(123,656)(124,652)(125,654)(126,653)(127,667)
(128,669)(129,668)(130,673)(131,675)(132,674)(133,670)(134,672)(135,671)
(136,685)(137,687)(138,686)(139,691)(140,693)(141,692)(142,688)(143,690)
(144,689)(145,676)(146,678)(147,677)(148,682)(149,684)(150,683)(151,679)
(152,681)(153,680)(154,694)(155,696)(156,695)(157,700)(158,702)(159,701)
(160,697)(161,699)(162,698)(163,712)(164,714)(165,713)(166,718)(167,720)
(168,719)(169,715)(170,717)(171,716)(172,703)(173,705)(174,704)(175,709)
(176,711)(177,710)(178,706)(179,708)(180,707)(181,721)(182,723)(183,722)
(184,727)(185,729)(186,728)(187,724)(188,726)(189,725)(190,739)(191,741)
(192,740)(193,745)(194,747)(195,746)(196,742)(197,744)(198,743)(199,730)
(200,732)(201,731)(202,736)(203,738)(204,737)(205,733)(206,735)(207,734)
(208,748)(209,750)(210,749)(211,754)(212,756)(213,755)(214,751)(215,753)
(216,752)(217,550)(218,552)(219,551)(220,556)(221,558)(222,557)(223,553)
(224,555)(225,554)(226,541)(227,543)(228,542)(229,547)(230,549)(231,548)
(232,544)(233,546)(234,545)(235,559)(236,561)(237,560)(238,565)(239,567)
(240,566)(241,562)(242,564)(243,563)(244,577)(245,579)(246,578)(247,583)
(248,585)(249,584)(250,580)(251,582)(252,581)(253,568)(254,570)(255,569)
(256,574)(257,576)(258,575)(259,571)(260,573)(261,572)(262,586)(263,588)
(264,587)(265,592)(266,594)(267,593)(268,589)(269,591)(270,590)(271,604)
(272,606)(273,605)(274,610)(275,612)(276,611)(277,607)(278,609)(279,608)
(280,595)(281,597)(282,596)(283,601)(284,603)(285,602)(286,598)(287,600)
(288,599)(289,613)(290,615)(291,614)(292,619)(293,621)(294,620)(295,616)
(296,618)(297,617)(298,631)(299,633)(300,632)(301,637)(302,639)(303,638)
(304,634)(305,636)(306,635)(307,622)(308,624)(309,623)(310,628)(311,630)
(312,629)(313,625)(314,627)(315,626)(316,640)(317,642)(318,641)(319,646)
(320,648)(321,647)(322,643)(323,645)(324,644)(325,442)(326,444)(327,443)
(328,448)(329,450)(330,449)(331,445)(332,447)(333,446)(334,433)(335,435)
(336,434)(337,439)(338,441)(339,440)(340,436)(341,438)(342,437)(343,451)
(344,453)(345,452)(346,457)(347,459)(348,458)(349,454)(350,456)(351,455)
(352,469)(353,471)(354,470)(355,475)(356,477)(357,476)(358,472)(359,474)
(360,473)(361,460)(362,462)(363,461)(364,466)(365,468)(366,467)(367,463)
(368,465)(369,464)(370,478)(371,480)(372,479)(373,484)(374,486)(375,485)
(376,481)(377,483)(378,482)(379,496)(380,498)(381,497)(382,502)(383,504)
(384,503)(385,499)(386,501)(387,500)(388,487)(389,489)(390,488)(391,493)
(392,495)(393,494)(394,490)(395,492)(396,491)(397,505)(398,507)(399,506)
(400,511)(401,513)(402,512)(403,508)(404,510)(405,509)(406,523)(407,525)
(408,524)(409,529)(410,531)(411,530)(412,526)(413,528)(414,527)(415,514)
(416,516)(417,515)(418,520)(419,522)(420,521)(421,517)(422,519)(423,518)
(424,532)(425,534)(426,533)(427,538)(428,540)(429,539)(430,535)(431,537)
(432,536);
s3 := Sym(864)!(  1,218)(  2,217)(  3,219)(  4,221)(  5,220)(  6,222)(  7,224)
(  8,223)(  9,225)( 10,227)( 11,226)( 12,228)( 13,230)( 14,229)( 15,231)
( 16,233)( 17,232)( 18,234)( 19,236)( 20,235)( 21,237)( 22,239)( 23,238)
( 24,240)( 25,242)( 26,241)( 27,243)( 28,245)( 29,244)( 30,246)( 31,248)
( 32,247)( 33,249)( 34,251)( 35,250)( 36,252)( 37,254)( 38,253)( 39,255)
( 40,257)( 41,256)( 42,258)( 43,260)( 44,259)( 45,261)( 46,263)( 47,262)
( 48,264)( 49,266)( 50,265)( 51,267)( 52,269)( 53,268)( 54,270)( 55,272)
( 56,271)( 57,273)( 58,275)( 59,274)( 60,276)( 61,278)( 62,277)( 63,279)
( 64,281)( 65,280)( 66,282)( 67,284)( 68,283)( 69,285)( 70,287)( 71,286)
( 72,288)( 73,290)( 74,289)( 75,291)( 76,293)( 77,292)( 78,294)( 79,296)
( 80,295)( 81,297)( 82,299)( 83,298)( 84,300)( 85,302)( 86,301)( 87,303)
( 88,305)( 89,304)( 90,306)( 91,308)( 92,307)( 93,309)( 94,311)( 95,310)
( 96,312)( 97,314)( 98,313)( 99,315)(100,317)(101,316)(102,318)(103,320)
(104,319)(105,321)(106,323)(107,322)(108,324)(109,380)(110,379)(111,381)
(112,383)(113,382)(114,384)(115,386)(116,385)(117,387)(118,389)(119,388)
(120,390)(121,392)(122,391)(123,393)(124,395)(125,394)(126,396)(127,398)
(128,397)(129,399)(130,401)(131,400)(132,402)(133,404)(134,403)(135,405)
(136,407)(137,406)(138,408)(139,410)(140,409)(141,411)(142,413)(143,412)
(144,414)(145,416)(146,415)(147,417)(148,419)(149,418)(150,420)(151,422)
(152,421)(153,423)(154,425)(155,424)(156,426)(157,428)(158,427)(159,429)
(160,431)(161,430)(162,432)(163,326)(164,325)(165,327)(166,329)(167,328)
(168,330)(169,332)(170,331)(171,333)(172,335)(173,334)(174,336)(175,338)
(176,337)(177,339)(178,341)(179,340)(180,342)(181,344)(182,343)(183,345)
(184,347)(185,346)(186,348)(187,350)(188,349)(189,351)(190,353)(191,352)
(192,354)(193,356)(194,355)(195,357)(196,359)(197,358)(198,360)(199,362)
(200,361)(201,363)(202,365)(203,364)(204,366)(205,368)(206,367)(207,369)
(208,371)(209,370)(210,372)(211,374)(212,373)(213,375)(214,377)(215,376)
(216,378)(433,650)(434,649)(435,651)(436,653)(437,652)(438,654)(439,656)
(440,655)(441,657)(442,659)(443,658)(444,660)(445,662)(446,661)(447,663)
(448,665)(449,664)(450,666)(451,668)(452,667)(453,669)(454,671)(455,670)
(456,672)(457,674)(458,673)(459,675)(460,677)(461,676)(462,678)(463,680)
(464,679)(465,681)(466,683)(467,682)(468,684)(469,686)(470,685)(471,687)
(472,689)(473,688)(474,690)(475,692)(476,691)(477,693)(478,695)(479,694)
(480,696)(481,698)(482,697)(483,699)(484,701)(485,700)(486,702)(487,704)
(488,703)(489,705)(490,707)(491,706)(492,708)(493,710)(494,709)(495,711)
(496,713)(497,712)(498,714)(499,716)(500,715)(501,717)(502,719)(503,718)
(504,720)(505,722)(506,721)(507,723)(508,725)(509,724)(510,726)(511,728)
(512,727)(513,729)(514,731)(515,730)(516,732)(517,734)(518,733)(519,735)
(520,737)(521,736)(522,738)(523,740)(524,739)(525,741)(526,743)(527,742)
(528,744)(529,746)(530,745)(531,747)(532,749)(533,748)(534,750)(535,752)
(536,751)(537,753)(538,755)(539,754)(540,756)(541,812)(542,811)(543,813)
(544,815)(545,814)(546,816)(547,818)(548,817)(549,819)(550,821)(551,820)
(552,822)(553,824)(554,823)(555,825)(556,827)(557,826)(558,828)(559,830)
(560,829)(561,831)(562,833)(563,832)(564,834)(565,836)(566,835)(567,837)
(568,839)(569,838)(570,840)(571,842)(572,841)(573,843)(574,845)(575,844)
(576,846)(577,848)(578,847)(579,849)(580,851)(581,850)(582,852)(583,854)
(584,853)(585,855)(586,857)(587,856)(588,858)(589,860)(590,859)(591,861)
(592,863)(593,862)(594,864)(595,758)(596,757)(597,759)(598,761)(599,760)
(600,762)(601,764)(602,763)(603,765)(604,767)(605,766)(606,768)(607,770)
(608,769)(609,771)(610,773)(611,772)(612,774)(613,776)(614,775)(615,777)
(616,779)(617,778)(618,780)(619,782)(620,781)(621,783)(622,785)(623,784)
(624,786)(625,788)(626,787)(627,789)(628,791)(629,790)(630,792)(631,794)
(632,793)(633,795)(634,797)(635,796)(636,798)(637,800)(638,799)(639,801)
(640,803)(641,802)(642,804)(643,806)(644,805)(645,807)(646,809)(647,808)
(648,810);
poly := sub<Sym(864)|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, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
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