Polytope of Type {12,6,9}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6,9}*1296b
if this polytope has a name.
Group : SmallGroup(1296,2020)
Rank : 4
Schlafli Type : {12,6,9}
Number of vertices, edges, etc : 12, 36, 27, 9
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 : {6,6,9}*648b
3-fold quotients : {12,2,9}*432, {4,6,9}*432, {12,6,3}*432b
6-fold quotients : {2,6,9}*216, {6,2,9}*216, {6,6,3}*216b
9-fold quotients : {4,2,9}*144, {12,2,3}*144, {4,6,3}*144
12-fold quotients : {3,2,9}*108
18-fold quotients : {2,2,9}*72, {2,6,3}*72, {6,2,3}*72
27-fold quotients : {4,2,3}*48
36-fold quotients : {3,2,3}*36
54-fold quotients : {2,2,3}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)( 23, 26)
( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)( 49, 52)
( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)( 69, 72)
( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)( 95, 98)
( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)(121,124)
(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)(141,144)
(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(163,244)(164,245)
(165,246)(166,250)(167,251)(168,252)(169,247)(170,248)(171,249)(172,253)
(173,254)(174,255)(175,259)(176,260)(177,261)(178,256)(179,257)(180,258)
(181,262)(182,263)(183,264)(184,268)(185,269)(186,270)(187,265)(188,266)
(189,267)(190,271)(191,272)(192,273)(193,277)(194,278)(195,279)(196,274)
(197,275)(198,276)(199,280)(200,281)(201,282)(202,286)(203,287)(204,288)
(205,283)(206,284)(207,285)(208,289)(209,290)(210,291)(211,295)(212,296)
(213,297)(214,292)(215,293)(216,294)(217,298)(218,299)(219,300)(220,304)
(221,305)(222,306)(223,301)(224,302)(225,303)(226,307)(227,308)(228,309)
(229,313)(230,314)(231,315)(232,310)(233,311)(234,312)(235,316)(236,317)
(237,318)(238,322)(239,323)(240,324)(241,319)(242,320)(243,321)(325,406)
(326,407)(327,408)(328,412)(329,413)(330,414)(331,409)(332,410)(333,411)
(334,415)(335,416)(336,417)(337,421)(338,422)(339,423)(340,418)(341,419)
(342,420)(343,424)(344,425)(345,426)(346,430)(347,431)(348,432)(349,427)
(350,428)(351,429)(352,433)(353,434)(354,435)(355,439)(356,440)(357,441)
(358,436)(359,437)(360,438)(361,442)(362,443)(363,444)(364,448)(365,449)
(366,450)(367,445)(368,446)(369,447)(370,451)(371,452)(372,453)(373,457)
(374,458)(375,459)(376,454)(377,455)(378,456)(379,460)(380,461)(381,462)
(382,466)(383,467)(384,468)(385,463)(386,464)(387,465)(388,469)(389,470)
(390,471)(391,475)(392,476)(393,477)(394,472)(395,473)(396,474)(397,478)
(398,479)(399,480)(400,484)(401,485)(402,486)(403,481)(404,482)(405,483)
(490,493)(491,494)(492,495)(499,502)(500,503)(501,504)(508,511)(509,512)
(510,513)(517,520)(518,521)(519,522)(526,529)(527,530)(528,531)(535,538)
(536,539)(537,540)(544,547)(545,548)(546,549)(553,556)(554,557)(555,558)
(562,565)(563,566)(564,567)(571,574)(572,575)(573,576)(580,583)(581,584)
(582,585)(589,592)(590,593)(591,594)(598,601)(599,602)(600,603)(607,610)
(608,611)(609,612)(616,619)(617,620)(618,621)(625,628)(626,629)(627,630)
(634,637)(635,638)(636,639)(643,646)(644,647)(645,648);;
s1 := ( 1,166)( 2,167)( 3,168)( 4,163)( 5,164)( 6,165)( 7,169)( 8,170)
( 9,171)( 10,184)( 11,185)( 12,186)( 13,181)( 14,182)( 15,183)( 16,187)
( 17,188)( 18,189)( 19,175)( 20,176)( 21,177)( 22,172)( 23,173)( 24,174)
( 25,178)( 26,179)( 27,180)( 28,193)( 29,194)( 30,195)( 31,190)( 32,191)
( 33,192)( 34,196)( 35,197)( 36,198)( 37,211)( 38,212)( 39,213)( 40,208)
( 41,209)( 42,210)( 43,214)( 44,215)( 45,216)( 46,202)( 47,203)( 48,204)
( 49,199)( 50,200)( 51,201)( 52,205)( 53,206)( 54,207)( 55,220)( 56,221)
( 57,222)( 58,217)( 59,218)( 60,219)( 61,223)( 62,224)( 63,225)( 64,238)
( 65,239)( 66,240)( 67,235)( 68,236)( 69,237)( 70,241)( 71,242)( 72,243)
( 73,229)( 74,230)( 75,231)( 76,226)( 77,227)( 78,228)( 79,232)( 80,233)
( 81,234)( 82,247)( 83,248)( 84,249)( 85,244)( 86,245)( 87,246)( 88,250)
( 89,251)( 90,252)( 91,265)( 92,266)( 93,267)( 94,262)( 95,263)( 96,264)
( 97,268)( 98,269)( 99,270)(100,256)(101,257)(102,258)(103,253)(104,254)
(105,255)(106,259)(107,260)(108,261)(109,274)(110,275)(111,276)(112,271)
(113,272)(114,273)(115,277)(116,278)(117,279)(118,292)(119,293)(120,294)
(121,289)(122,290)(123,291)(124,295)(125,296)(126,297)(127,283)(128,284)
(129,285)(130,280)(131,281)(132,282)(133,286)(134,287)(135,288)(136,301)
(137,302)(138,303)(139,298)(140,299)(141,300)(142,304)(143,305)(144,306)
(145,319)(146,320)(147,321)(148,316)(149,317)(150,318)(151,322)(152,323)
(153,324)(154,310)(155,311)(156,312)(157,307)(158,308)(159,309)(160,313)
(161,314)(162,315)(325,571)(326,572)(327,573)(328,568)(329,569)(330,570)
(331,574)(332,575)(333,576)(334,589)(335,590)(336,591)(337,586)(338,587)
(339,588)(340,592)(341,593)(342,594)(343,580)(344,581)(345,582)(346,577)
(347,578)(348,579)(349,583)(350,584)(351,585)(352,598)(353,599)(354,600)
(355,595)(356,596)(357,597)(358,601)(359,602)(360,603)(361,616)(362,617)
(363,618)(364,613)(365,614)(366,615)(367,619)(368,620)(369,621)(370,607)
(371,608)(372,609)(373,604)(374,605)(375,606)(376,610)(377,611)(378,612)
(379,625)(380,626)(381,627)(382,622)(383,623)(384,624)(385,628)(386,629)
(387,630)(388,643)(389,644)(390,645)(391,640)(392,641)(393,642)(394,646)
(395,647)(396,648)(397,634)(398,635)(399,636)(400,631)(401,632)(402,633)
(403,637)(404,638)(405,639)(406,490)(407,491)(408,492)(409,487)(410,488)
(411,489)(412,493)(413,494)(414,495)(415,508)(416,509)(417,510)(418,505)
(419,506)(420,507)(421,511)(422,512)(423,513)(424,499)(425,500)(426,501)
(427,496)(428,497)(429,498)(430,502)(431,503)(432,504)(433,517)(434,518)
(435,519)(436,514)(437,515)(438,516)(439,520)(440,521)(441,522)(442,535)
(443,536)(444,537)(445,532)(446,533)(447,534)(448,538)(449,539)(450,540)
(451,526)(452,527)(453,528)(454,523)(455,524)(456,525)(457,529)(458,530)
(459,531)(460,544)(461,545)(462,546)(463,541)(464,542)(465,543)(466,547)
(467,548)(468,549)(469,562)(470,563)(471,564)(472,559)(473,560)(474,561)
(475,565)(476,566)(477,567)(478,553)(479,554)(480,555)(481,550)(482,551)
(483,552)(484,556)(485,557)(486,558);;
s2 := ( 1,496)( 2,498)( 3,497)( 4,499)( 5,501)( 6,500)( 7,502)( 8,504)
( 9,503)( 10,487)( 11,489)( 12,488)( 13,490)( 14,492)( 15,491)( 16,493)
( 17,495)( 18,494)( 19,505)( 20,507)( 21,506)( 22,508)( 23,510)( 24,509)
( 25,511)( 26,513)( 27,512)( 28,552)( 29,551)( 30,550)( 31,555)( 32,554)
( 33,553)( 34,558)( 35,557)( 36,556)( 37,543)( 38,542)( 39,541)( 40,546)
( 41,545)( 42,544)( 43,549)( 44,548)( 45,547)( 46,561)( 47,560)( 48,559)
( 49,564)( 50,563)( 51,562)( 52,567)( 53,566)( 54,565)( 55,525)( 56,524)
( 57,523)( 58,528)( 59,527)( 60,526)( 61,531)( 62,530)( 63,529)( 64,516)
( 65,515)( 66,514)( 67,519)( 68,518)( 69,517)( 70,522)( 71,521)( 72,520)
( 73,534)( 74,533)( 75,532)( 76,537)( 77,536)( 78,535)( 79,540)( 80,539)
( 81,538)( 82,577)( 83,579)( 84,578)( 85,580)( 86,582)( 87,581)( 88,583)
( 89,585)( 90,584)( 91,568)( 92,570)( 93,569)( 94,571)( 95,573)( 96,572)
( 97,574)( 98,576)( 99,575)(100,586)(101,588)(102,587)(103,589)(104,591)
(105,590)(106,592)(107,594)(108,593)(109,633)(110,632)(111,631)(112,636)
(113,635)(114,634)(115,639)(116,638)(117,637)(118,624)(119,623)(120,622)
(121,627)(122,626)(123,625)(124,630)(125,629)(126,628)(127,642)(128,641)
(129,640)(130,645)(131,644)(132,643)(133,648)(134,647)(135,646)(136,606)
(137,605)(138,604)(139,609)(140,608)(141,607)(142,612)(143,611)(144,610)
(145,597)(146,596)(147,595)(148,600)(149,599)(150,598)(151,603)(152,602)
(153,601)(154,615)(155,614)(156,613)(157,618)(158,617)(159,616)(160,621)
(161,620)(162,619)(163,415)(164,417)(165,416)(166,418)(167,420)(168,419)
(169,421)(170,423)(171,422)(172,406)(173,408)(174,407)(175,409)(176,411)
(177,410)(178,412)(179,414)(180,413)(181,424)(182,426)(183,425)(184,427)
(185,429)(186,428)(187,430)(188,432)(189,431)(190,471)(191,470)(192,469)
(193,474)(194,473)(195,472)(196,477)(197,476)(198,475)(199,462)(200,461)
(201,460)(202,465)(203,464)(204,463)(205,468)(206,467)(207,466)(208,480)
(209,479)(210,478)(211,483)(212,482)(213,481)(214,486)(215,485)(216,484)
(217,444)(218,443)(219,442)(220,447)(221,446)(222,445)(223,450)(224,449)
(225,448)(226,435)(227,434)(228,433)(229,438)(230,437)(231,436)(232,441)
(233,440)(234,439)(235,453)(236,452)(237,451)(238,456)(239,455)(240,454)
(241,459)(242,458)(243,457)(244,334)(245,336)(246,335)(247,337)(248,339)
(249,338)(250,340)(251,342)(252,341)(253,325)(254,327)(255,326)(256,328)
(257,330)(258,329)(259,331)(260,333)(261,332)(262,343)(263,345)(264,344)
(265,346)(266,348)(267,347)(268,349)(269,351)(270,350)(271,390)(272,389)
(273,388)(274,393)(275,392)(276,391)(277,396)(278,395)(279,394)(280,381)
(281,380)(282,379)(283,384)(284,383)(285,382)(286,387)(287,386)(288,385)
(289,399)(290,398)(291,397)(292,402)(293,401)(294,400)(295,405)(296,404)
(297,403)(298,363)(299,362)(300,361)(301,366)(302,365)(303,364)(304,369)
(305,368)(306,367)(307,354)(308,353)(309,352)(310,357)(311,356)(312,355)
(313,360)(314,359)(315,358)(316,372)(317,371)(318,370)(319,375)(320,374)
(321,373)(322,378)(323,377)(324,376);;
s3 := ( 1,514)( 2,516)( 3,515)( 4,517)( 5,519)( 6,518)( 7,520)( 8,522)
( 9,521)( 10,532)( 11,534)( 12,533)( 13,535)( 14,537)( 15,536)( 16,538)
( 17,540)( 18,539)( 19,523)( 20,525)( 21,524)( 22,526)( 23,528)( 24,527)
( 25,529)( 26,531)( 27,530)( 28,487)( 29,489)( 30,488)( 31,490)( 32,492)
( 33,491)( 34,493)( 35,495)( 36,494)( 37,505)( 38,507)( 39,506)( 40,508)
( 41,510)( 42,509)( 43,511)( 44,513)( 45,512)( 46,496)( 47,498)( 48,497)
( 49,499)( 50,501)( 51,500)( 52,502)( 53,504)( 54,503)( 55,543)( 56,542)
( 57,541)( 58,546)( 59,545)( 60,544)( 61,549)( 62,548)( 63,547)( 64,561)
( 65,560)( 66,559)( 67,564)( 68,563)( 69,562)( 70,567)( 71,566)( 72,565)
( 73,552)( 74,551)( 75,550)( 76,555)( 77,554)( 78,553)( 79,558)( 80,557)
( 81,556)( 82,595)( 83,597)( 84,596)( 85,598)( 86,600)( 87,599)( 88,601)
( 89,603)( 90,602)( 91,613)( 92,615)( 93,614)( 94,616)( 95,618)( 96,617)
( 97,619)( 98,621)( 99,620)(100,604)(101,606)(102,605)(103,607)(104,609)
(105,608)(106,610)(107,612)(108,611)(109,568)(110,570)(111,569)(112,571)
(113,573)(114,572)(115,574)(116,576)(117,575)(118,586)(119,588)(120,587)
(121,589)(122,591)(123,590)(124,592)(125,594)(126,593)(127,577)(128,579)
(129,578)(130,580)(131,582)(132,581)(133,583)(134,585)(135,584)(136,624)
(137,623)(138,622)(139,627)(140,626)(141,625)(142,630)(143,629)(144,628)
(145,642)(146,641)(147,640)(148,645)(149,644)(150,643)(151,648)(152,647)
(153,646)(154,633)(155,632)(156,631)(157,636)(158,635)(159,634)(160,639)
(161,638)(162,637)(163,433)(164,435)(165,434)(166,436)(167,438)(168,437)
(169,439)(170,441)(171,440)(172,451)(173,453)(174,452)(175,454)(176,456)
(177,455)(178,457)(179,459)(180,458)(181,442)(182,444)(183,443)(184,445)
(185,447)(186,446)(187,448)(188,450)(189,449)(190,406)(191,408)(192,407)
(193,409)(194,411)(195,410)(196,412)(197,414)(198,413)(199,424)(200,426)
(201,425)(202,427)(203,429)(204,428)(205,430)(206,432)(207,431)(208,415)
(209,417)(210,416)(211,418)(212,420)(213,419)(214,421)(215,423)(216,422)
(217,462)(218,461)(219,460)(220,465)(221,464)(222,463)(223,468)(224,467)
(225,466)(226,480)(227,479)(228,478)(229,483)(230,482)(231,481)(232,486)
(233,485)(234,484)(235,471)(236,470)(237,469)(238,474)(239,473)(240,472)
(241,477)(242,476)(243,475)(244,352)(245,354)(246,353)(247,355)(248,357)
(249,356)(250,358)(251,360)(252,359)(253,370)(254,372)(255,371)(256,373)
(257,375)(258,374)(259,376)(260,378)(261,377)(262,361)(263,363)(264,362)
(265,364)(266,366)(267,365)(268,367)(269,369)(270,368)(271,325)(272,327)
(273,326)(274,328)(275,330)(276,329)(277,331)(278,333)(279,332)(280,343)
(281,345)(282,344)(283,346)(284,348)(285,347)(286,349)(287,351)(288,350)
(289,334)(290,336)(291,335)(292,337)(293,339)(294,338)(295,340)(296,342)
(297,341)(298,381)(299,380)(300,379)(301,384)(302,383)(303,382)(304,387)
(305,386)(306,385)(307,399)(308,398)(309,397)(310,402)(311,401)(312,400)
(313,405)(314,404)(315,403)(316,390)(317,389)(318,388)(319,393)(320,392)
(321,391)(322,396)(323,395)(324,394);;
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*s1*s2*s3*s1*s2*s1*s2, 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,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(648)!( 4, 7)( 5, 8)( 6, 9)( 13, 16)( 14, 17)( 15, 18)( 22, 25)
( 23, 26)( 24, 27)( 31, 34)( 32, 35)( 33, 36)( 40, 43)( 41, 44)( 42, 45)
( 49, 52)( 50, 53)( 51, 54)( 58, 61)( 59, 62)( 60, 63)( 67, 70)( 68, 71)
( 69, 72)( 76, 79)( 77, 80)( 78, 81)( 85, 88)( 86, 89)( 87, 90)( 94, 97)
( 95, 98)( 96, 99)(103,106)(104,107)(105,108)(112,115)(113,116)(114,117)
(121,124)(122,125)(123,126)(130,133)(131,134)(132,135)(139,142)(140,143)
(141,144)(148,151)(149,152)(150,153)(157,160)(158,161)(159,162)(163,244)
(164,245)(165,246)(166,250)(167,251)(168,252)(169,247)(170,248)(171,249)
(172,253)(173,254)(174,255)(175,259)(176,260)(177,261)(178,256)(179,257)
(180,258)(181,262)(182,263)(183,264)(184,268)(185,269)(186,270)(187,265)
(188,266)(189,267)(190,271)(191,272)(192,273)(193,277)(194,278)(195,279)
(196,274)(197,275)(198,276)(199,280)(200,281)(201,282)(202,286)(203,287)
(204,288)(205,283)(206,284)(207,285)(208,289)(209,290)(210,291)(211,295)
(212,296)(213,297)(214,292)(215,293)(216,294)(217,298)(218,299)(219,300)
(220,304)(221,305)(222,306)(223,301)(224,302)(225,303)(226,307)(227,308)
(228,309)(229,313)(230,314)(231,315)(232,310)(233,311)(234,312)(235,316)
(236,317)(237,318)(238,322)(239,323)(240,324)(241,319)(242,320)(243,321)
(325,406)(326,407)(327,408)(328,412)(329,413)(330,414)(331,409)(332,410)
(333,411)(334,415)(335,416)(336,417)(337,421)(338,422)(339,423)(340,418)
(341,419)(342,420)(343,424)(344,425)(345,426)(346,430)(347,431)(348,432)
(349,427)(350,428)(351,429)(352,433)(353,434)(354,435)(355,439)(356,440)
(357,441)(358,436)(359,437)(360,438)(361,442)(362,443)(363,444)(364,448)
(365,449)(366,450)(367,445)(368,446)(369,447)(370,451)(371,452)(372,453)
(373,457)(374,458)(375,459)(376,454)(377,455)(378,456)(379,460)(380,461)
(381,462)(382,466)(383,467)(384,468)(385,463)(386,464)(387,465)(388,469)
(389,470)(390,471)(391,475)(392,476)(393,477)(394,472)(395,473)(396,474)
(397,478)(398,479)(399,480)(400,484)(401,485)(402,486)(403,481)(404,482)
(405,483)(490,493)(491,494)(492,495)(499,502)(500,503)(501,504)(508,511)
(509,512)(510,513)(517,520)(518,521)(519,522)(526,529)(527,530)(528,531)
(535,538)(536,539)(537,540)(544,547)(545,548)(546,549)(553,556)(554,557)
(555,558)(562,565)(563,566)(564,567)(571,574)(572,575)(573,576)(580,583)
(581,584)(582,585)(589,592)(590,593)(591,594)(598,601)(599,602)(600,603)
(607,610)(608,611)(609,612)(616,619)(617,620)(618,621)(625,628)(626,629)
(627,630)(634,637)(635,638)(636,639)(643,646)(644,647)(645,648);
s1 := Sym(648)!( 1,166)( 2,167)( 3,168)( 4,163)( 5,164)( 6,165)( 7,169)
( 8,170)( 9,171)( 10,184)( 11,185)( 12,186)( 13,181)( 14,182)( 15,183)
( 16,187)( 17,188)( 18,189)( 19,175)( 20,176)( 21,177)( 22,172)( 23,173)
( 24,174)( 25,178)( 26,179)( 27,180)( 28,193)( 29,194)( 30,195)( 31,190)
( 32,191)( 33,192)( 34,196)( 35,197)( 36,198)( 37,211)( 38,212)( 39,213)
( 40,208)( 41,209)( 42,210)( 43,214)( 44,215)( 45,216)( 46,202)( 47,203)
( 48,204)( 49,199)( 50,200)( 51,201)( 52,205)( 53,206)( 54,207)( 55,220)
( 56,221)( 57,222)( 58,217)( 59,218)( 60,219)( 61,223)( 62,224)( 63,225)
( 64,238)( 65,239)( 66,240)( 67,235)( 68,236)( 69,237)( 70,241)( 71,242)
( 72,243)( 73,229)( 74,230)( 75,231)( 76,226)( 77,227)( 78,228)( 79,232)
( 80,233)( 81,234)( 82,247)( 83,248)( 84,249)( 85,244)( 86,245)( 87,246)
( 88,250)( 89,251)( 90,252)( 91,265)( 92,266)( 93,267)( 94,262)( 95,263)
( 96,264)( 97,268)( 98,269)( 99,270)(100,256)(101,257)(102,258)(103,253)
(104,254)(105,255)(106,259)(107,260)(108,261)(109,274)(110,275)(111,276)
(112,271)(113,272)(114,273)(115,277)(116,278)(117,279)(118,292)(119,293)
(120,294)(121,289)(122,290)(123,291)(124,295)(125,296)(126,297)(127,283)
(128,284)(129,285)(130,280)(131,281)(132,282)(133,286)(134,287)(135,288)
(136,301)(137,302)(138,303)(139,298)(140,299)(141,300)(142,304)(143,305)
(144,306)(145,319)(146,320)(147,321)(148,316)(149,317)(150,318)(151,322)
(152,323)(153,324)(154,310)(155,311)(156,312)(157,307)(158,308)(159,309)
(160,313)(161,314)(162,315)(325,571)(326,572)(327,573)(328,568)(329,569)
(330,570)(331,574)(332,575)(333,576)(334,589)(335,590)(336,591)(337,586)
(338,587)(339,588)(340,592)(341,593)(342,594)(343,580)(344,581)(345,582)
(346,577)(347,578)(348,579)(349,583)(350,584)(351,585)(352,598)(353,599)
(354,600)(355,595)(356,596)(357,597)(358,601)(359,602)(360,603)(361,616)
(362,617)(363,618)(364,613)(365,614)(366,615)(367,619)(368,620)(369,621)
(370,607)(371,608)(372,609)(373,604)(374,605)(375,606)(376,610)(377,611)
(378,612)(379,625)(380,626)(381,627)(382,622)(383,623)(384,624)(385,628)
(386,629)(387,630)(388,643)(389,644)(390,645)(391,640)(392,641)(393,642)
(394,646)(395,647)(396,648)(397,634)(398,635)(399,636)(400,631)(401,632)
(402,633)(403,637)(404,638)(405,639)(406,490)(407,491)(408,492)(409,487)
(410,488)(411,489)(412,493)(413,494)(414,495)(415,508)(416,509)(417,510)
(418,505)(419,506)(420,507)(421,511)(422,512)(423,513)(424,499)(425,500)
(426,501)(427,496)(428,497)(429,498)(430,502)(431,503)(432,504)(433,517)
(434,518)(435,519)(436,514)(437,515)(438,516)(439,520)(440,521)(441,522)
(442,535)(443,536)(444,537)(445,532)(446,533)(447,534)(448,538)(449,539)
(450,540)(451,526)(452,527)(453,528)(454,523)(455,524)(456,525)(457,529)
(458,530)(459,531)(460,544)(461,545)(462,546)(463,541)(464,542)(465,543)
(466,547)(467,548)(468,549)(469,562)(470,563)(471,564)(472,559)(473,560)
(474,561)(475,565)(476,566)(477,567)(478,553)(479,554)(480,555)(481,550)
(482,551)(483,552)(484,556)(485,557)(486,558);
s2 := Sym(648)!( 1,496)( 2,498)( 3,497)( 4,499)( 5,501)( 6,500)( 7,502)
( 8,504)( 9,503)( 10,487)( 11,489)( 12,488)( 13,490)( 14,492)( 15,491)
( 16,493)( 17,495)( 18,494)( 19,505)( 20,507)( 21,506)( 22,508)( 23,510)
( 24,509)( 25,511)( 26,513)( 27,512)( 28,552)( 29,551)( 30,550)( 31,555)
( 32,554)( 33,553)( 34,558)( 35,557)( 36,556)( 37,543)( 38,542)( 39,541)
( 40,546)( 41,545)( 42,544)( 43,549)( 44,548)( 45,547)( 46,561)( 47,560)
( 48,559)( 49,564)( 50,563)( 51,562)( 52,567)( 53,566)( 54,565)( 55,525)
( 56,524)( 57,523)( 58,528)( 59,527)( 60,526)( 61,531)( 62,530)( 63,529)
( 64,516)( 65,515)( 66,514)( 67,519)( 68,518)( 69,517)( 70,522)( 71,521)
( 72,520)( 73,534)( 74,533)( 75,532)( 76,537)( 77,536)( 78,535)( 79,540)
( 80,539)( 81,538)( 82,577)( 83,579)( 84,578)( 85,580)( 86,582)( 87,581)
( 88,583)( 89,585)( 90,584)( 91,568)( 92,570)( 93,569)( 94,571)( 95,573)
( 96,572)( 97,574)( 98,576)( 99,575)(100,586)(101,588)(102,587)(103,589)
(104,591)(105,590)(106,592)(107,594)(108,593)(109,633)(110,632)(111,631)
(112,636)(113,635)(114,634)(115,639)(116,638)(117,637)(118,624)(119,623)
(120,622)(121,627)(122,626)(123,625)(124,630)(125,629)(126,628)(127,642)
(128,641)(129,640)(130,645)(131,644)(132,643)(133,648)(134,647)(135,646)
(136,606)(137,605)(138,604)(139,609)(140,608)(141,607)(142,612)(143,611)
(144,610)(145,597)(146,596)(147,595)(148,600)(149,599)(150,598)(151,603)
(152,602)(153,601)(154,615)(155,614)(156,613)(157,618)(158,617)(159,616)
(160,621)(161,620)(162,619)(163,415)(164,417)(165,416)(166,418)(167,420)
(168,419)(169,421)(170,423)(171,422)(172,406)(173,408)(174,407)(175,409)
(176,411)(177,410)(178,412)(179,414)(180,413)(181,424)(182,426)(183,425)
(184,427)(185,429)(186,428)(187,430)(188,432)(189,431)(190,471)(191,470)
(192,469)(193,474)(194,473)(195,472)(196,477)(197,476)(198,475)(199,462)
(200,461)(201,460)(202,465)(203,464)(204,463)(205,468)(206,467)(207,466)
(208,480)(209,479)(210,478)(211,483)(212,482)(213,481)(214,486)(215,485)
(216,484)(217,444)(218,443)(219,442)(220,447)(221,446)(222,445)(223,450)
(224,449)(225,448)(226,435)(227,434)(228,433)(229,438)(230,437)(231,436)
(232,441)(233,440)(234,439)(235,453)(236,452)(237,451)(238,456)(239,455)
(240,454)(241,459)(242,458)(243,457)(244,334)(245,336)(246,335)(247,337)
(248,339)(249,338)(250,340)(251,342)(252,341)(253,325)(254,327)(255,326)
(256,328)(257,330)(258,329)(259,331)(260,333)(261,332)(262,343)(263,345)
(264,344)(265,346)(266,348)(267,347)(268,349)(269,351)(270,350)(271,390)
(272,389)(273,388)(274,393)(275,392)(276,391)(277,396)(278,395)(279,394)
(280,381)(281,380)(282,379)(283,384)(284,383)(285,382)(286,387)(287,386)
(288,385)(289,399)(290,398)(291,397)(292,402)(293,401)(294,400)(295,405)
(296,404)(297,403)(298,363)(299,362)(300,361)(301,366)(302,365)(303,364)
(304,369)(305,368)(306,367)(307,354)(308,353)(309,352)(310,357)(311,356)
(312,355)(313,360)(314,359)(315,358)(316,372)(317,371)(318,370)(319,375)
(320,374)(321,373)(322,378)(323,377)(324,376);
s3 := Sym(648)!( 1,514)( 2,516)( 3,515)( 4,517)( 5,519)( 6,518)( 7,520)
( 8,522)( 9,521)( 10,532)( 11,534)( 12,533)( 13,535)( 14,537)( 15,536)
( 16,538)( 17,540)( 18,539)( 19,523)( 20,525)( 21,524)( 22,526)( 23,528)
( 24,527)( 25,529)( 26,531)( 27,530)( 28,487)( 29,489)( 30,488)( 31,490)
( 32,492)( 33,491)( 34,493)( 35,495)( 36,494)( 37,505)( 38,507)( 39,506)
( 40,508)( 41,510)( 42,509)( 43,511)( 44,513)( 45,512)( 46,496)( 47,498)
( 48,497)( 49,499)( 50,501)( 51,500)( 52,502)( 53,504)( 54,503)( 55,543)
( 56,542)( 57,541)( 58,546)( 59,545)( 60,544)( 61,549)( 62,548)( 63,547)
( 64,561)( 65,560)( 66,559)( 67,564)( 68,563)( 69,562)( 70,567)( 71,566)
( 72,565)( 73,552)( 74,551)( 75,550)( 76,555)( 77,554)( 78,553)( 79,558)
( 80,557)( 81,556)( 82,595)( 83,597)( 84,596)( 85,598)( 86,600)( 87,599)
( 88,601)( 89,603)( 90,602)( 91,613)( 92,615)( 93,614)( 94,616)( 95,618)
( 96,617)( 97,619)( 98,621)( 99,620)(100,604)(101,606)(102,605)(103,607)
(104,609)(105,608)(106,610)(107,612)(108,611)(109,568)(110,570)(111,569)
(112,571)(113,573)(114,572)(115,574)(116,576)(117,575)(118,586)(119,588)
(120,587)(121,589)(122,591)(123,590)(124,592)(125,594)(126,593)(127,577)
(128,579)(129,578)(130,580)(131,582)(132,581)(133,583)(134,585)(135,584)
(136,624)(137,623)(138,622)(139,627)(140,626)(141,625)(142,630)(143,629)
(144,628)(145,642)(146,641)(147,640)(148,645)(149,644)(150,643)(151,648)
(152,647)(153,646)(154,633)(155,632)(156,631)(157,636)(158,635)(159,634)
(160,639)(161,638)(162,637)(163,433)(164,435)(165,434)(166,436)(167,438)
(168,437)(169,439)(170,441)(171,440)(172,451)(173,453)(174,452)(175,454)
(176,456)(177,455)(178,457)(179,459)(180,458)(181,442)(182,444)(183,443)
(184,445)(185,447)(186,446)(187,448)(188,450)(189,449)(190,406)(191,408)
(192,407)(193,409)(194,411)(195,410)(196,412)(197,414)(198,413)(199,424)
(200,426)(201,425)(202,427)(203,429)(204,428)(205,430)(206,432)(207,431)
(208,415)(209,417)(210,416)(211,418)(212,420)(213,419)(214,421)(215,423)
(216,422)(217,462)(218,461)(219,460)(220,465)(221,464)(222,463)(223,468)
(224,467)(225,466)(226,480)(227,479)(228,478)(229,483)(230,482)(231,481)
(232,486)(233,485)(234,484)(235,471)(236,470)(237,469)(238,474)(239,473)
(240,472)(241,477)(242,476)(243,475)(244,352)(245,354)(246,353)(247,355)
(248,357)(249,356)(250,358)(251,360)(252,359)(253,370)(254,372)(255,371)
(256,373)(257,375)(258,374)(259,376)(260,378)(261,377)(262,361)(263,363)
(264,362)(265,364)(266,366)(267,365)(268,367)(269,369)(270,368)(271,325)
(272,327)(273,326)(274,328)(275,330)(276,329)(277,331)(278,333)(279,332)
(280,343)(281,345)(282,344)(283,346)(284,348)(285,347)(286,349)(287,351)
(288,350)(289,334)(290,336)(291,335)(292,337)(293,339)(294,338)(295,340)
(296,342)(297,341)(298,381)(299,380)(300,379)(301,384)(302,383)(303,382)
(304,387)(305,386)(306,385)(307,399)(308,398)(309,397)(310,402)(311,401)
(312,400)(313,405)(314,404)(315,403)(316,390)(317,389)(318,388)(319,393)
(320,392)(321,391)(322,396)(323,395)(324,394);
poly := sub<Sym(648)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s2*s1*s0*s1*s2*s1, s3*s1*s2*s1*s2*s3*s1*s2*s1*s2,
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,
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