Polytope of Type {12,12}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,12}*864h
if this polytope has a name.
Group : SmallGroup(864,2919)
Rank : 3
Schlafli Type : {12,12}
Number of vertices, edges, etc : 36, 216, 36
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Dual
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {12,12,2} of size 1728
Vertex Figure Of :
   {2,12,12} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,12}*432g, {12,6}*432g
   3-fold quotients : {12,12}*288a, {12,12}*288b, {12,12}*288c
   4-fold quotients : {6,6}*216d
   6-fold quotients : {6,12}*144a, {6,12}*144b, {12,6}*144a, {12,6}*144b, {6,12}*144c, {12,6}*144c
   9-fold quotients : {4,12}*96a, {12,4}*96a
   12-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   18-fold quotients : {2,12}*48, {12,2}*48, {4,6}*48a, {6,4}*48a
   24-fold quotients : {3,6}*36, {6,3}*36
   27-fold quotients : {4,4}*32
   36-fold quotients : {2,6}*24, {6,2}*24
   54-fold quotients : {2,4}*16, {4,2}*16
   72-fold quotients : {2,3}*12, {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {12,24}*1728o, {24,12}*1728o, {12,24}*1728p, {24,12}*1728p, {12,12}*1728h
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s0*s1, 
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