Polytope of Type {4,18,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,18,12}*1728b
if this polytope has a name.
Group : SmallGroup(1728,17257)
Rank : 4
Schlafli Type : {4,18,12}
Number of vertices, edges, etc : 4, 36, 108, 12
Order of s0s1s2s3 : 36
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 : {4,18,6}*864b, {2,18,12}*864b
3-fold quotients : {4,18,4}*576a, {4,6,12}*576c
4-fold quotients : {2,18,6}*432b
6-fold quotients : {2,18,4}*288a, {4,18,2}*288a, {4,6,6}*288b, {2,6,12}*288c
8-fold quotients : {2,9,6}*216
9-fold quotients : {4,6,4}*192a
12-fold quotients : {2,18,2}*144, {2,6,6}*144c
18-fold quotients : {2,6,4}*96a, {4,6,2}*96a
24-fold quotients : {2,9,2}*72, {2,3,6}*72
27-fold quotients : {4,2,4}*64
36-fold quotients : {2,6,2}*48
54-fold quotients : {2,2,4}*32, {4,2,2}*32
72-fold quotients : {2,3,2}*24
108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,217)( 2,218)( 3,219)( 4,220)( 5,221)( 6,222)( 7,223)( 8,224)
( 9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)( 16,232)
( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)( 24,240)
( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)( 32,248)
( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)( 40,256)
( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)( 48,264)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)( 56,272)
( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)( 64,280)
( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)( 72,288)
( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)( 80,296)
( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)( 88,304)
( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)( 96,312)
( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)(104,320)
(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)(112,382)
(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)(120,390)
(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)(128,398)
(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)(136,406)
(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)(144,414)
(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)(152,422)
(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)
(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)
(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)(176,338)
(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)(184,346)
(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)(192,354)
(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)(200,362)
(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)(208,370)
(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)(216,378)
(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)(440,656)
(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)(448,664)
(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)(456,672)
(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)(464,680)
(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)(472,688)
(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)(480,696)
(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)(488,704)
(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)(496,712)
(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)(504,720)
(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)(512,728)
(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)(520,736)
(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)(528,744)
(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)(536,752)
(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)(544,814)
(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)(552,822)
(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)(560,830)
(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)(568,838)
(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)(576,846)
(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)(584,854)
(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)(592,862)
(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)(600,762)
(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)(608,770)
(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)(616,778)
(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)(624,786)
(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)(632,794)
(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)(640,802)
(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)(648,810);;
s1 := ( 1,541)( 2,543)( 3,542)( 4,549)( 5,548)( 6,547)( 7,546)( 8,545)
( 9,544)( 10,559)( 11,561)( 12,560)( 13,567)( 14,566)( 15,565)( 16,564)
( 17,563)( 18,562)( 19,550)( 20,552)( 21,551)( 22,558)( 23,557)( 24,556)
( 25,555)( 26,554)( 27,553)( 28,568)( 29,570)( 30,569)( 31,576)( 32,575)
( 33,574)( 34,573)( 35,572)( 36,571)( 37,586)( 38,588)( 39,587)( 40,594)
( 41,593)( 42,592)( 43,591)( 44,590)( 45,589)( 46,577)( 47,579)( 48,578)
( 49,585)( 50,584)( 51,583)( 52,582)( 53,581)( 54,580)( 55,595)( 56,597)
( 57,596)( 58,603)( 59,602)( 60,601)( 61,600)( 62,599)( 63,598)( 64,613)
( 65,615)( 66,614)( 67,621)( 68,620)( 69,619)( 70,618)( 71,617)( 72,616)
( 73,604)( 74,606)( 75,605)( 76,612)( 77,611)( 78,610)( 79,609)( 80,608)
( 81,607)( 82,622)( 83,624)( 84,623)( 85,630)( 86,629)( 87,628)( 88,627)
( 89,626)( 90,625)( 91,640)( 92,642)( 93,641)( 94,648)( 95,647)( 96,646)
( 97,645)( 98,644)( 99,643)(100,631)(101,633)(102,632)(103,639)(104,638)
(105,637)(106,636)(107,635)(108,634)(109,433)(110,435)(111,434)(112,441)
(113,440)(114,439)(115,438)(116,437)(117,436)(118,451)(119,453)(120,452)
(121,459)(122,458)(123,457)(124,456)(125,455)(126,454)(127,442)(128,444)
(129,443)(130,450)(131,449)(132,448)(133,447)(134,446)(135,445)(136,460)
(137,462)(138,461)(139,468)(140,467)(141,466)(142,465)(143,464)(144,463)
(145,478)(146,480)(147,479)(148,486)(149,485)(150,484)(151,483)(152,482)
(153,481)(154,469)(155,471)(156,470)(157,477)(158,476)(159,475)(160,474)
(161,473)(162,472)(163,487)(164,489)(165,488)(166,495)(167,494)(168,493)
(169,492)(170,491)(171,490)(172,505)(173,507)(174,506)(175,513)(176,512)
(177,511)(178,510)(179,509)(180,508)(181,496)(182,498)(183,497)(184,504)
(185,503)(186,502)(187,501)(188,500)(189,499)(190,514)(191,516)(192,515)
(193,522)(194,521)(195,520)(196,519)(197,518)(198,517)(199,532)(200,534)
(201,533)(202,540)(203,539)(204,538)(205,537)(206,536)(207,535)(208,523)
(209,525)(210,524)(211,531)(212,530)(213,529)(214,528)(215,527)(216,526)
(217,757)(218,759)(219,758)(220,765)(221,764)(222,763)(223,762)(224,761)
(225,760)(226,775)(227,777)(228,776)(229,783)(230,782)(231,781)(232,780)
(233,779)(234,778)(235,766)(236,768)(237,767)(238,774)(239,773)(240,772)
(241,771)(242,770)(243,769)(244,784)(245,786)(246,785)(247,792)(248,791)
(249,790)(250,789)(251,788)(252,787)(253,802)(254,804)(255,803)(256,810)
(257,809)(258,808)(259,807)(260,806)(261,805)(262,793)(263,795)(264,794)
(265,801)(266,800)(267,799)(268,798)(269,797)(270,796)(271,811)(272,813)
(273,812)(274,819)(275,818)(276,817)(277,816)(278,815)(279,814)(280,829)
(281,831)(282,830)(283,837)(284,836)(285,835)(286,834)(287,833)(288,832)
(289,820)(290,822)(291,821)(292,828)(293,827)(294,826)(295,825)(296,824)
(297,823)(298,838)(299,840)(300,839)(301,846)(302,845)(303,844)(304,843)
(305,842)(306,841)(307,856)(308,858)(309,857)(310,864)(311,863)(312,862)
(313,861)(314,860)(315,859)(316,847)(317,849)(318,848)(319,855)(320,854)
(321,853)(322,852)(323,851)(324,850)(325,649)(326,651)(327,650)(328,657)
(329,656)(330,655)(331,654)(332,653)(333,652)(334,667)(335,669)(336,668)
(337,675)(338,674)(339,673)(340,672)(341,671)(342,670)(343,658)(344,660)
(345,659)(346,666)(347,665)(348,664)(349,663)(350,662)(351,661)(352,676)
(353,678)(354,677)(355,684)(356,683)(357,682)(358,681)(359,680)(360,679)
(361,694)(362,696)(363,695)(364,702)(365,701)(366,700)(367,699)(368,698)
(369,697)(370,685)(371,687)(372,686)(373,693)(374,692)(375,691)(376,690)
(377,689)(378,688)(379,703)(380,705)(381,704)(382,711)(383,710)(384,709)
(385,708)(386,707)(387,706)(388,721)(389,723)(390,722)(391,729)(392,728)
(393,727)(394,726)(395,725)(396,724)(397,712)(398,714)(399,713)(400,720)
(401,719)(402,718)(403,717)(404,716)(405,715)(406,730)(407,732)(408,731)
(409,738)(410,737)(411,736)(412,735)(413,734)(414,733)(415,748)(416,750)
(417,749)(418,756)(419,755)(420,754)(421,753)(422,752)(423,751)(424,739)
(425,741)(426,740)(427,747)(428,746)(429,745)(430,744)(431,743)(432,742);;
s2 := ( 1,445)( 2,447)( 3,446)( 4,442)( 5,444)( 6,443)( 7,450)( 8,449)
( 9,448)( 10,436)( 11,438)( 12,437)( 13,433)( 14,435)( 15,434)( 16,441)
( 17,440)( 18,439)( 19,454)( 20,456)( 21,455)( 22,451)( 23,453)( 24,452)
( 25,459)( 26,458)( 27,457)( 28,472)( 29,474)( 30,473)( 31,469)( 32,471)
( 33,470)( 34,477)( 35,476)( 36,475)( 37,463)( 38,465)( 39,464)( 40,460)
( 41,462)( 42,461)( 43,468)( 44,467)( 45,466)( 46,481)( 47,483)( 48,482)
( 49,478)( 50,480)( 51,479)( 52,486)( 53,485)( 54,484)( 55,499)( 56,501)
( 57,500)( 58,496)( 59,498)( 60,497)( 61,504)( 62,503)( 63,502)( 64,490)
( 65,492)( 66,491)( 67,487)( 68,489)( 69,488)( 70,495)( 71,494)( 72,493)
( 73,508)( 74,510)( 75,509)( 76,505)( 77,507)( 78,506)( 79,513)( 80,512)
( 81,511)( 82,526)( 83,528)( 84,527)( 85,523)( 86,525)( 87,524)( 88,531)
( 89,530)( 90,529)( 91,517)( 92,519)( 93,518)( 94,514)( 95,516)( 96,515)
( 97,522)( 98,521)( 99,520)(100,535)(101,537)(102,536)(103,532)(104,534)
(105,533)(106,540)(107,539)(108,538)(109,553)(110,555)(111,554)(112,550)
(113,552)(114,551)(115,558)(116,557)(117,556)(118,544)(119,546)(120,545)
(121,541)(122,543)(123,542)(124,549)(125,548)(126,547)(127,562)(128,564)
(129,563)(130,559)(131,561)(132,560)(133,567)(134,566)(135,565)(136,580)
(137,582)(138,581)(139,577)(140,579)(141,578)(142,585)(143,584)(144,583)
(145,571)(146,573)(147,572)(148,568)(149,570)(150,569)(151,576)(152,575)
(153,574)(154,589)(155,591)(156,590)(157,586)(158,588)(159,587)(160,594)
(161,593)(162,592)(163,607)(164,609)(165,608)(166,604)(167,606)(168,605)
(169,612)(170,611)(171,610)(172,598)(173,600)(174,599)(175,595)(176,597)
(177,596)(178,603)(179,602)(180,601)(181,616)(182,618)(183,617)(184,613)
(185,615)(186,614)(187,621)(188,620)(189,619)(190,634)(191,636)(192,635)
(193,631)(194,633)(195,632)(196,639)(197,638)(198,637)(199,625)(200,627)
(201,626)(202,622)(203,624)(204,623)(205,630)(206,629)(207,628)(208,643)
(209,645)(210,644)(211,640)(212,642)(213,641)(214,648)(215,647)(216,646)
(217,661)(218,663)(219,662)(220,658)(221,660)(222,659)(223,666)(224,665)
(225,664)(226,652)(227,654)(228,653)(229,649)(230,651)(231,650)(232,657)
(233,656)(234,655)(235,670)(236,672)(237,671)(238,667)(239,669)(240,668)
(241,675)(242,674)(243,673)(244,688)(245,690)(246,689)(247,685)(248,687)
(249,686)(250,693)(251,692)(252,691)(253,679)(254,681)(255,680)(256,676)
(257,678)(258,677)(259,684)(260,683)(261,682)(262,697)(263,699)(264,698)
(265,694)(266,696)(267,695)(268,702)(269,701)(270,700)(271,715)(272,717)
(273,716)(274,712)(275,714)(276,713)(277,720)(278,719)(279,718)(280,706)
(281,708)(282,707)(283,703)(284,705)(285,704)(286,711)(287,710)(288,709)
(289,724)(290,726)(291,725)(292,721)(293,723)(294,722)(295,729)(296,728)
(297,727)(298,742)(299,744)(300,743)(301,739)(302,741)(303,740)(304,747)
(305,746)(306,745)(307,733)(308,735)(309,734)(310,730)(311,732)(312,731)
(313,738)(314,737)(315,736)(316,751)(317,753)(318,752)(319,748)(320,750)
(321,749)(322,756)(323,755)(324,754)(325,769)(326,771)(327,770)(328,766)
(329,768)(330,767)(331,774)(332,773)(333,772)(334,760)(335,762)(336,761)
(337,757)(338,759)(339,758)(340,765)(341,764)(342,763)(343,778)(344,780)
(345,779)(346,775)(347,777)(348,776)(349,783)(350,782)(351,781)(352,796)
(353,798)(354,797)(355,793)(356,795)(357,794)(358,801)(359,800)(360,799)
(361,787)(362,789)(363,788)(364,784)(365,786)(366,785)(367,792)(368,791)
(369,790)(370,805)(371,807)(372,806)(373,802)(374,804)(375,803)(376,810)
(377,809)(378,808)(379,823)(380,825)(381,824)(382,820)(383,822)(384,821)
(385,828)(386,827)(387,826)(388,814)(389,816)(390,815)(391,811)(392,813)
(393,812)(394,819)(395,818)(396,817)(397,832)(398,834)(399,833)(400,829)
(401,831)(402,830)(403,837)(404,836)(405,835)(406,850)(407,852)(408,851)
(409,847)(410,849)(411,848)(412,855)(413,854)(414,853)(415,841)(416,843)
(417,842)(418,838)(419,840)(420,839)(421,846)(422,845)(423,844)(424,859)
(425,861)(426,860)(427,856)(428,858)(429,857)(430,864)(431,863)(432,862);;
s3 := ( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)( 17, 26)
( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)( 43, 52)
( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)( 69, 78)
( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)( 95,104)
( 96,105)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)(112,139)
(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)(120,156)
(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)(128,146)
(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)(163,190)
(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)(171,198)
(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)(179,215)
(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)(187,205)
(188,206)(189,207)(226,235)(227,236)(228,237)(229,238)(230,239)(231,240)
(232,241)(233,242)(234,243)(253,262)(254,263)(255,264)(256,265)(257,266)
(258,267)(259,268)(260,269)(261,270)(280,289)(281,290)(282,291)(283,292)
(284,293)(285,294)(286,295)(287,296)(288,297)(307,316)(308,317)(309,318)
(310,319)(311,320)(312,321)(313,322)(314,323)(315,324)(325,352)(326,353)
(327,354)(328,355)(329,356)(330,357)(331,358)(332,359)(333,360)(334,370)
(335,371)(336,372)(337,373)(338,374)(339,375)(340,376)(341,377)(342,378)
(343,361)(344,362)(345,363)(346,364)(347,365)(348,366)(349,367)(350,368)
(351,369)(379,406)(380,407)(381,408)(382,409)(383,410)(384,411)(385,412)
(386,413)(387,414)(388,424)(389,425)(390,426)(391,427)(392,428)(393,429)
(394,430)(395,431)(396,432)(397,415)(398,416)(399,417)(400,418)(401,419)
(402,420)(403,421)(404,422)(405,423)(433,460)(434,461)(435,462)(436,463)
(437,464)(438,465)(439,466)(440,467)(441,468)(442,478)(443,479)(444,480)
(445,481)(446,482)(447,483)(448,484)(449,485)(450,486)(451,469)(452,470)
(453,471)(454,472)(455,473)(456,474)(457,475)(458,476)(459,477)(487,514)
(488,515)(489,516)(490,517)(491,518)(492,519)(493,520)(494,521)(495,522)
(496,532)(497,533)(498,534)(499,535)(500,536)(501,537)(502,538)(503,539)
(504,540)(505,523)(506,524)(507,525)(508,526)(509,527)(510,528)(511,529)
(512,530)(513,531)(550,559)(551,560)(552,561)(553,562)(554,563)(555,564)
(556,565)(557,566)(558,567)(577,586)(578,587)(579,588)(580,589)(581,590)
(582,591)(583,592)(584,593)(585,594)(604,613)(605,614)(606,615)(607,616)
(608,617)(609,618)(610,619)(611,620)(612,621)(631,640)(632,641)(633,642)
(634,643)(635,644)(636,645)(637,646)(638,647)(639,648)(649,676)(650,677)
(651,678)(652,679)(653,680)(654,681)(655,682)(656,683)(657,684)(658,694)
(659,695)(660,696)(661,697)(662,698)(663,699)(664,700)(665,701)(666,702)
(667,685)(668,686)(669,687)(670,688)(671,689)(672,690)(673,691)(674,692)
(675,693)(703,730)(704,731)(705,732)(706,733)(707,734)(708,735)(709,736)
(710,737)(711,738)(712,748)(713,749)(714,750)(715,751)(716,752)(717,753)
(718,754)(719,755)(720,756)(721,739)(722,740)(723,741)(724,742)(725,743)
(726,744)(727,745)(728,746)(729,747)(766,775)(767,776)(768,777)(769,778)
(770,779)(771,780)(772,781)(773,782)(774,783)(793,802)(794,803)(795,804)
(796,805)(797,806)(798,807)(799,808)(800,809)(801,810)(820,829)(821,830)
(822,831)(823,832)(824,833)(825,834)(826,835)(827,836)(828,837)(847,856)
(848,857)(849,858)(850,859)(851,860)(852,861)(853,862)(854,863)(855,864);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s3*s2*s3*s1*s2*s3*s2,
s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(864)!( 1,217)( 2,218)( 3,219)( 4,220)( 5,221)( 6,222)( 7,223)
( 8,224)( 9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)
( 16,232)( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)
( 24,240)( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)
( 32,248)( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)
( 40,256)( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)
( 48,264)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,271)
( 56,272)( 57,273)( 58,274)( 59,275)( 60,276)( 61,277)( 62,278)( 63,279)
( 64,280)( 65,281)( 66,282)( 67,283)( 68,284)( 69,285)( 70,286)( 71,287)
( 72,288)( 73,289)( 74,290)( 75,291)( 76,292)( 77,293)( 78,294)( 79,295)
( 80,296)( 81,297)( 82,298)( 83,299)( 84,300)( 85,301)( 86,302)( 87,303)
( 88,304)( 89,305)( 90,306)( 91,307)( 92,308)( 93,309)( 94,310)( 95,311)
( 96,312)( 97,313)( 98,314)( 99,315)(100,316)(101,317)(102,318)(103,319)
(104,320)(105,321)(106,322)(107,323)(108,324)(109,379)(110,380)(111,381)
(112,382)(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)
(120,390)(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)
(128,398)(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)
(136,406)(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)
(144,414)(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)
(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)
(160,430)(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)
(168,330)(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)
(176,338)(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)
(184,346)(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)
(192,354)(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)
(200,362)(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)
(208,370)(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)
(216,378)(433,649)(434,650)(435,651)(436,652)(437,653)(438,654)(439,655)
(440,656)(441,657)(442,658)(443,659)(444,660)(445,661)(446,662)(447,663)
(448,664)(449,665)(450,666)(451,667)(452,668)(453,669)(454,670)(455,671)
(456,672)(457,673)(458,674)(459,675)(460,676)(461,677)(462,678)(463,679)
(464,680)(465,681)(466,682)(467,683)(468,684)(469,685)(470,686)(471,687)
(472,688)(473,689)(474,690)(475,691)(476,692)(477,693)(478,694)(479,695)
(480,696)(481,697)(482,698)(483,699)(484,700)(485,701)(486,702)(487,703)
(488,704)(489,705)(490,706)(491,707)(492,708)(493,709)(494,710)(495,711)
(496,712)(497,713)(498,714)(499,715)(500,716)(501,717)(502,718)(503,719)
(504,720)(505,721)(506,722)(507,723)(508,724)(509,725)(510,726)(511,727)
(512,728)(513,729)(514,730)(515,731)(516,732)(517,733)(518,734)(519,735)
(520,736)(521,737)(522,738)(523,739)(524,740)(525,741)(526,742)(527,743)
(528,744)(529,745)(530,746)(531,747)(532,748)(533,749)(534,750)(535,751)
(536,752)(537,753)(538,754)(539,755)(540,756)(541,811)(542,812)(543,813)
(544,814)(545,815)(546,816)(547,817)(548,818)(549,819)(550,820)(551,821)
(552,822)(553,823)(554,824)(555,825)(556,826)(557,827)(558,828)(559,829)
(560,830)(561,831)(562,832)(563,833)(564,834)(565,835)(566,836)(567,837)
(568,838)(569,839)(570,840)(571,841)(572,842)(573,843)(574,844)(575,845)
(576,846)(577,847)(578,848)(579,849)(580,850)(581,851)(582,852)(583,853)
(584,854)(585,855)(586,856)(587,857)(588,858)(589,859)(590,860)(591,861)
(592,862)(593,863)(594,864)(595,757)(596,758)(597,759)(598,760)(599,761)
(600,762)(601,763)(602,764)(603,765)(604,766)(605,767)(606,768)(607,769)
(608,770)(609,771)(610,772)(611,773)(612,774)(613,775)(614,776)(615,777)
(616,778)(617,779)(618,780)(619,781)(620,782)(621,783)(622,784)(623,785)
(624,786)(625,787)(626,788)(627,789)(628,790)(629,791)(630,792)(631,793)
(632,794)(633,795)(634,796)(635,797)(636,798)(637,799)(638,800)(639,801)
(640,802)(641,803)(642,804)(643,805)(644,806)(645,807)(646,808)(647,809)
(648,810);
s1 := Sym(864)!( 1,541)( 2,543)( 3,542)( 4,549)( 5,548)( 6,547)( 7,546)
( 8,545)( 9,544)( 10,559)( 11,561)( 12,560)( 13,567)( 14,566)( 15,565)
( 16,564)( 17,563)( 18,562)( 19,550)( 20,552)( 21,551)( 22,558)( 23,557)
( 24,556)( 25,555)( 26,554)( 27,553)( 28,568)( 29,570)( 30,569)( 31,576)
( 32,575)( 33,574)( 34,573)( 35,572)( 36,571)( 37,586)( 38,588)( 39,587)
( 40,594)( 41,593)( 42,592)( 43,591)( 44,590)( 45,589)( 46,577)( 47,579)
( 48,578)( 49,585)( 50,584)( 51,583)( 52,582)( 53,581)( 54,580)( 55,595)
( 56,597)( 57,596)( 58,603)( 59,602)( 60,601)( 61,600)( 62,599)( 63,598)
( 64,613)( 65,615)( 66,614)( 67,621)( 68,620)( 69,619)( 70,618)( 71,617)
( 72,616)( 73,604)( 74,606)( 75,605)( 76,612)( 77,611)( 78,610)( 79,609)
( 80,608)( 81,607)( 82,622)( 83,624)( 84,623)( 85,630)( 86,629)( 87,628)
( 88,627)( 89,626)( 90,625)( 91,640)( 92,642)( 93,641)( 94,648)( 95,647)
( 96,646)( 97,645)( 98,644)( 99,643)(100,631)(101,633)(102,632)(103,639)
(104,638)(105,637)(106,636)(107,635)(108,634)(109,433)(110,435)(111,434)
(112,441)(113,440)(114,439)(115,438)(116,437)(117,436)(118,451)(119,453)
(120,452)(121,459)(122,458)(123,457)(124,456)(125,455)(126,454)(127,442)
(128,444)(129,443)(130,450)(131,449)(132,448)(133,447)(134,446)(135,445)
(136,460)(137,462)(138,461)(139,468)(140,467)(141,466)(142,465)(143,464)
(144,463)(145,478)(146,480)(147,479)(148,486)(149,485)(150,484)(151,483)
(152,482)(153,481)(154,469)(155,471)(156,470)(157,477)(158,476)(159,475)
(160,474)(161,473)(162,472)(163,487)(164,489)(165,488)(166,495)(167,494)
(168,493)(169,492)(170,491)(171,490)(172,505)(173,507)(174,506)(175,513)
(176,512)(177,511)(178,510)(179,509)(180,508)(181,496)(182,498)(183,497)
(184,504)(185,503)(186,502)(187,501)(188,500)(189,499)(190,514)(191,516)
(192,515)(193,522)(194,521)(195,520)(196,519)(197,518)(198,517)(199,532)
(200,534)(201,533)(202,540)(203,539)(204,538)(205,537)(206,536)(207,535)
(208,523)(209,525)(210,524)(211,531)(212,530)(213,529)(214,528)(215,527)
(216,526)(217,757)(218,759)(219,758)(220,765)(221,764)(222,763)(223,762)
(224,761)(225,760)(226,775)(227,777)(228,776)(229,783)(230,782)(231,781)
(232,780)(233,779)(234,778)(235,766)(236,768)(237,767)(238,774)(239,773)
(240,772)(241,771)(242,770)(243,769)(244,784)(245,786)(246,785)(247,792)
(248,791)(249,790)(250,789)(251,788)(252,787)(253,802)(254,804)(255,803)
(256,810)(257,809)(258,808)(259,807)(260,806)(261,805)(262,793)(263,795)
(264,794)(265,801)(266,800)(267,799)(268,798)(269,797)(270,796)(271,811)
(272,813)(273,812)(274,819)(275,818)(276,817)(277,816)(278,815)(279,814)
(280,829)(281,831)(282,830)(283,837)(284,836)(285,835)(286,834)(287,833)
(288,832)(289,820)(290,822)(291,821)(292,828)(293,827)(294,826)(295,825)
(296,824)(297,823)(298,838)(299,840)(300,839)(301,846)(302,845)(303,844)
(304,843)(305,842)(306,841)(307,856)(308,858)(309,857)(310,864)(311,863)
(312,862)(313,861)(314,860)(315,859)(316,847)(317,849)(318,848)(319,855)
(320,854)(321,853)(322,852)(323,851)(324,850)(325,649)(326,651)(327,650)
(328,657)(329,656)(330,655)(331,654)(332,653)(333,652)(334,667)(335,669)
(336,668)(337,675)(338,674)(339,673)(340,672)(341,671)(342,670)(343,658)
(344,660)(345,659)(346,666)(347,665)(348,664)(349,663)(350,662)(351,661)
(352,676)(353,678)(354,677)(355,684)(356,683)(357,682)(358,681)(359,680)
(360,679)(361,694)(362,696)(363,695)(364,702)(365,701)(366,700)(367,699)
(368,698)(369,697)(370,685)(371,687)(372,686)(373,693)(374,692)(375,691)
(376,690)(377,689)(378,688)(379,703)(380,705)(381,704)(382,711)(383,710)
(384,709)(385,708)(386,707)(387,706)(388,721)(389,723)(390,722)(391,729)
(392,728)(393,727)(394,726)(395,725)(396,724)(397,712)(398,714)(399,713)
(400,720)(401,719)(402,718)(403,717)(404,716)(405,715)(406,730)(407,732)
(408,731)(409,738)(410,737)(411,736)(412,735)(413,734)(414,733)(415,748)
(416,750)(417,749)(418,756)(419,755)(420,754)(421,753)(422,752)(423,751)
(424,739)(425,741)(426,740)(427,747)(428,746)(429,745)(430,744)(431,743)
(432,742);
s2 := Sym(864)!( 1,445)( 2,447)( 3,446)( 4,442)( 5,444)( 6,443)( 7,450)
( 8,449)( 9,448)( 10,436)( 11,438)( 12,437)( 13,433)( 14,435)( 15,434)
( 16,441)( 17,440)( 18,439)( 19,454)( 20,456)( 21,455)( 22,451)( 23,453)
( 24,452)( 25,459)( 26,458)( 27,457)( 28,472)( 29,474)( 30,473)( 31,469)
( 32,471)( 33,470)( 34,477)( 35,476)( 36,475)( 37,463)( 38,465)( 39,464)
( 40,460)( 41,462)( 42,461)( 43,468)( 44,467)( 45,466)( 46,481)( 47,483)
( 48,482)( 49,478)( 50,480)( 51,479)( 52,486)( 53,485)( 54,484)( 55,499)
( 56,501)( 57,500)( 58,496)( 59,498)( 60,497)( 61,504)( 62,503)( 63,502)
( 64,490)( 65,492)( 66,491)( 67,487)( 68,489)( 69,488)( 70,495)( 71,494)
( 72,493)( 73,508)( 74,510)( 75,509)( 76,505)( 77,507)( 78,506)( 79,513)
( 80,512)( 81,511)( 82,526)( 83,528)( 84,527)( 85,523)( 86,525)( 87,524)
( 88,531)( 89,530)( 90,529)( 91,517)( 92,519)( 93,518)( 94,514)( 95,516)
( 96,515)( 97,522)( 98,521)( 99,520)(100,535)(101,537)(102,536)(103,532)
(104,534)(105,533)(106,540)(107,539)(108,538)(109,553)(110,555)(111,554)
(112,550)(113,552)(114,551)(115,558)(116,557)(117,556)(118,544)(119,546)
(120,545)(121,541)(122,543)(123,542)(124,549)(125,548)(126,547)(127,562)
(128,564)(129,563)(130,559)(131,561)(132,560)(133,567)(134,566)(135,565)
(136,580)(137,582)(138,581)(139,577)(140,579)(141,578)(142,585)(143,584)
(144,583)(145,571)(146,573)(147,572)(148,568)(149,570)(150,569)(151,576)
(152,575)(153,574)(154,589)(155,591)(156,590)(157,586)(158,588)(159,587)
(160,594)(161,593)(162,592)(163,607)(164,609)(165,608)(166,604)(167,606)
(168,605)(169,612)(170,611)(171,610)(172,598)(173,600)(174,599)(175,595)
(176,597)(177,596)(178,603)(179,602)(180,601)(181,616)(182,618)(183,617)
(184,613)(185,615)(186,614)(187,621)(188,620)(189,619)(190,634)(191,636)
(192,635)(193,631)(194,633)(195,632)(196,639)(197,638)(198,637)(199,625)
(200,627)(201,626)(202,622)(203,624)(204,623)(205,630)(206,629)(207,628)
(208,643)(209,645)(210,644)(211,640)(212,642)(213,641)(214,648)(215,647)
(216,646)(217,661)(218,663)(219,662)(220,658)(221,660)(222,659)(223,666)
(224,665)(225,664)(226,652)(227,654)(228,653)(229,649)(230,651)(231,650)
(232,657)(233,656)(234,655)(235,670)(236,672)(237,671)(238,667)(239,669)
(240,668)(241,675)(242,674)(243,673)(244,688)(245,690)(246,689)(247,685)
(248,687)(249,686)(250,693)(251,692)(252,691)(253,679)(254,681)(255,680)
(256,676)(257,678)(258,677)(259,684)(260,683)(261,682)(262,697)(263,699)
(264,698)(265,694)(266,696)(267,695)(268,702)(269,701)(270,700)(271,715)
(272,717)(273,716)(274,712)(275,714)(276,713)(277,720)(278,719)(279,718)
(280,706)(281,708)(282,707)(283,703)(284,705)(285,704)(286,711)(287,710)
(288,709)(289,724)(290,726)(291,725)(292,721)(293,723)(294,722)(295,729)
(296,728)(297,727)(298,742)(299,744)(300,743)(301,739)(302,741)(303,740)
(304,747)(305,746)(306,745)(307,733)(308,735)(309,734)(310,730)(311,732)
(312,731)(313,738)(314,737)(315,736)(316,751)(317,753)(318,752)(319,748)
(320,750)(321,749)(322,756)(323,755)(324,754)(325,769)(326,771)(327,770)
(328,766)(329,768)(330,767)(331,774)(332,773)(333,772)(334,760)(335,762)
(336,761)(337,757)(338,759)(339,758)(340,765)(341,764)(342,763)(343,778)
(344,780)(345,779)(346,775)(347,777)(348,776)(349,783)(350,782)(351,781)
(352,796)(353,798)(354,797)(355,793)(356,795)(357,794)(358,801)(359,800)
(360,799)(361,787)(362,789)(363,788)(364,784)(365,786)(366,785)(367,792)
(368,791)(369,790)(370,805)(371,807)(372,806)(373,802)(374,804)(375,803)
(376,810)(377,809)(378,808)(379,823)(380,825)(381,824)(382,820)(383,822)
(384,821)(385,828)(386,827)(387,826)(388,814)(389,816)(390,815)(391,811)
(392,813)(393,812)(394,819)(395,818)(396,817)(397,832)(398,834)(399,833)
(400,829)(401,831)(402,830)(403,837)(404,836)(405,835)(406,850)(407,852)
(408,851)(409,847)(410,849)(411,848)(412,855)(413,854)(414,853)(415,841)
(416,843)(417,842)(418,838)(419,840)(420,839)(421,846)(422,845)(423,844)
(424,859)(425,861)(426,860)(427,856)(428,858)(429,857)(430,864)(431,863)
(432,862);
s3 := Sym(864)!( 10, 19)( 11, 20)( 12, 21)( 13, 22)( 14, 23)( 15, 24)( 16, 25)
( 17, 26)( 18, 27)( 37, 46)( 38, 47)( 39, 48)( 40, 49)( 41, 50)( 42, 51)
( 43, 52)( 44, 53)( 45, 54)( 64, 73)( 65, 74)( 66, 75)( 67, 76)( 68, 77)
( 69, 78)( 70, 79)( 71, 80)( 72, 81)( 91,100)( 92,101)( 93,102)( 94,103)
( 95,104)( 96,105)( 97,106)( 98,107)( 99,108)(109,136)(110,137)(111,138)
(112,139)(113,140)(114,141)(115,142)(116,143)(117,144)(118,154)(119,155)
(120,156)(121,157)(122,158)(123,159)(124,160)(125,161)(126,162)(127,145)
(128,146)(129,147)(130,148)(131,149)(132,150)(133,151)(134,152)(135,153)
(163,190)(164,191)(165,192)(166,193)(167,194)(168,195)(169,196)(170,197)
(171,198)(172,208)(173,209)(174,210)(175,211)(176,212)(177,213)(178,214)
(179,215)(180,216)(181,199)(182,200)(183,201)(184,202)(185,203)(186,204)
(187,205)(188,206)(189,207)(226,235)(227,236)(228,237)(229,238)(230,239)
(231,240)(232,241)(233,242)(234,243)(253,262)(254,263)(255,264)(256,265)
(257,266)(258,267)(259,268)(260,269)(261,270)(280,289)(281,290)(282,291)
(283,292)(284,293)(285,294)(286,295)(287,296)(288,297)(307,316)(308,317)
(309,318)(310,319)(311,320)(312,321)(313,322)(314,323)(315,324)(325,352)
(326,353)(327,354)(328,355)(329,356)(330,357)(331,358)(332,359)(333,360)
(334,370)(335,371)(336,372)(337,373)(338,374)(339,375)(340,376)(341,377)
(342,378)(343,361)(344,362)(345,363)(346,364)(347,365)(348,366)(349,367)
(350,368)(351,369)(379,406)(380,407)(381,408)(382,409)(383,410)(384,411)
(385,412)(386,413)(387,414)(388,424)(389,425)(390,426)(391,427)(392,428)
(393,429)(394,430)(395,431)(396,432)(397,415)(398,416)(399,417)(400,418)
(401,419)(402,420)(403,421)(404,422)(405,423)(433,460)(434,461)(435,462)
(436,463)(437,464)(438,465)(439,466)(440,467)(441,468)(442,478)(443,479)
(444,480)(445,481)(446,482)(447,483)(448,484)(449,485)(450,486)(451,469)
(452,470)(453,471)(454,472)(455,473)(456,474)(457,475)(458,476)(459,477)
(487,514)(488,515)(489,516)(490,517)(491,518)(492,519)(493,520)(494,521)
(495,522)(496,532)(497,533)(498,534)(499,535)(500,536)(501,537)(502,538)
(503,539)(504,540)(505,523)(506,524)(507,525)(508,526)(509,527)(510,528)
(511,529)(512,530)(513,531)(550,559)(551,560)(552,561)(553,562)(554,563)
(555,564)(556,565)(557,566)(558,567)(577,586)(578,587)(579,588)(580,589)
(581,590)(582,591)(583,592)(584,593)(585,594)(604,613)(605,614)(606,615)
(607,616)(608,617)(609,618)(610,619)(611,620)(612,621)(631,640)(632,641)
(633,642)(634,643)(635,644)(636,645)(637,646)(638,647)(639,648)(649,676)
(650,677)(651,678)(652,679)(653,680)(654,681)(655,682)(656,683)(657,684)
(658,694)(659,695)(660,696)(661,697)(662,698)(663,699)(664,700)(665,701)
(666,702)(667,685)(668,686)(669,687)(670,688)(671,689)(672,690)(673,691)
(674,692)(675,693)(703,730)(704,731)(705,732)(706,733)(707,734)(708,735)
(709,736)(710,737)(711,738)(712,748)(713,749)(714,750)(715,751)(716,752)
(717,753)(718,754)(719,755)(720,756)(721,739)(722,740)(723,741)(724,742)
(725,743)(726,744)(727,745)(728,746)(729,747)(766,775)(767,776)(768,777)
(769,778)(770,779)(771,780)(772,781)(773,782)(774,783)(793,802)(794,803)
(795,804)(796,805)(797,806)(798,807)(799,808)(800,809)(801,810)(820,829)
(821,830)(822,831)(823,832)(824,833)(825,834)(826,835)(827,836)(828,837)
(847,856)(848,857)(849,858)(850,859)(851,860)(852,861)(853,862)(854,863)
(855,864);
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*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s3*s1*s2*s3*s2*s3*s1*s2*s3*s2, s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s3*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >;
References : None.
to this polytope