Questions?
See the FAQ
or other info.

Polytope of Type {12,6,12}

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