Polytope of Type {3,6,12}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {3,6,12}*1296c
if this polytope has a name.
Group : SmallGroup(1296,2061)
Rank : 4
Schlafli Type : {3,6,12}
Number of vertices, edges, etc : 3, 27, 108, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 6
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 : {3,6,6}*648e
3-fold quotients : {3,6,12}*432a, {3,6,12}*432b
6-fold quotients : {3,6,6}*216a, {3,6,6}*216b
9-fold quotients : {3,2,12}*144, {3,6,4}*144
12-fold quotients : {3,6,3}*108
18-fold quotients : {3,2,6}*72, {3,6,2}*72
27-fold quotients : {3,2,4}*48
36-fold quotients : {3,2,3}*36
54-fold quotients : {3,2,2}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,487)( 2,489)( 3,488)( 4,490)( 5,492)( 6,491)( 7,493)( 8,495)
( 9,494)( 10,505)( 11,507)( 12,506)( 13,508)( 14,510)( 15,509)( 16,511)
( 17,513)( 18,512)( 19,496)( 20,498)( 21,497)( 22,499)( 23,501)( 24,500)
( 25,502)( 26,504)( 27,503)( 28,541)( 29,543)( 30,542)( 31,544)( 32,546)
( 33,545)( 34,547)( 35,549)( 36,548)( 37,559)( 38,561)( 39,560)( 40,562)
( 41,564)( 42,563)( 43,565)( 44,567)( 45,566)( 46,550)( 47,552)( 48,551)
( 49,553)( 50,555)( 51,554)( 52,556)( 53,558)( 54,557)( 55,514)( 56,516)
( 57,515)( 58,517)( 59,519)( 60,518)( 61,520)( 62,522)( 63,521)( 64,532)
( 65,534)( 66,533)( 67,535)( 68,537)( 69,536)( 70,538)( 71,540)( 72,539)
( 73,523)( 74,525)( 75,524)( 76,526)( 77,528)( 78,527)( 79,529)( 80,531)
( 81,530)( 82,568)( 83,570)( 84,569)( 85,571)( 86,573)( 87,572)( 88,574)
( 89,576)( 90,575)( 91,586)( 92,588)( 93,587)( 94,589)( 95,591)( 96,590)
( 97,592)( 98,594)( 99,593)(100,577)(101,579)(102,578)(103,580)(104,582)
(105,581)(106,583)(107,585)(108,584)(109,622)(110,624)(111,623)(112,625)
(113,627)(114,626)(115,628)(116,630)(117,629)(118,640)(119,642)(120,641)
(121,643)(122,645)(123,644)(124,646)(125,648)(126,647)(127,631)(128,633)
(129,632)(130,634)(131,636)(132,635)(133,637)(134,639)(135,638)(136,595)
(137,597)(138,596)(139,598)(140,600)(141,599)(142,601)(143,603)(144,602)
(145,613)(146,615)(147,614)(148,616)(149,618)(150,617)(151,619)(152,621)
(153,620)(154,604)(155,606)(156,605)(157,607)(158,609)(159,608)(160,610)
(161,612)(162,611)(163,406)(164,408)(165,407)(166,409)(167,411)(168,410)
(169,412)(170,414)(171,413)(172,424)(173,426)(174,425)(175,427)(176,429)
(177,428)(178,430)(179,432)(180,431)(181,415)(182,417)(183,416)(184,418)
(185,420)(186,419)(187,421)(188,423)(189,422)(190,460)(191,462)(192,461)
(193,463)(194,465)(195,464)(196,466)(197,468)(198,467)(199,478)(200,480)
(201,479)(202,481)(203,483)(204,482)(205,484)(206,486)(207,485)(208,469)
(209,471)(210,470)(211,472)(212,474)(213,473)(214,475)(215,477)(216,476)
(217,433)(218,435)(219,434)(220,436)(221,438)(222,437)(223,439)(224,441)
(225,440)(226,451)(227,453)(228,452)(229,454)(230,456)(231,455)(232,457)
(233,459)(234,458)(235,442)(236,444)(237,443)(238,445)(239,447)(240,446)
(241,448)(242,450)(243,449)(244,325)(245,327)(246,326)(247,328)(248,330)
(249,329)(250,331)(251,333)(252,332)(253,343)(254,345)(255,344)(256,346)
(257,348)(258,347)(259,349)(260,351)(261,350)(262,334)(263,336)(264,335)
(265,337)(266,339)(267,338)(268,340)(269,342)(270,341)(271,379)(272,381)
(273,380)(274,382)(275,384)(276,383)(277,385)(278,387)(279,386)(280,397)
(281,399)(282,398)(283,400)(284,402)(285,401)(286,403)(287,405)(288,404)
(289,388)(290,390)(291,389)(292,391)(293,393)(294,392)(295,394)(296,396)
(297,395)(298,352)(299,354)(300,353)(301,355)(302,357)(303,356)(304,358)
(305,360)(306,359)(307,370)(308,372)(309,371)(310,373)(311,375)(312,374)
(313,376)(314,378)(315,377)(316,361)(317,363)(318,362)(319,364)(320,366)
(321,365)(322,367)(323,369)(324,368);;
s1 := ( 1,523)( 2,525)( 3,524)( 4,527)( 5,526)( 6,528)( 7,531)( 8,530)
( 9,529)( 10,514)( 11,516)( 12,515)( 13,518)( 14,517)( 15,519)( 16,522)
( 17,521)( 18,520)( 19,532)( 20,534)( 21,533)( 22,536)( 23,535)( 24,537)
( 25,540)( 26,539)( 27,538)( 28,496)( 29,498)( 30,497)( 31,500)( 32,499)
( 33,501)( 34,504)( 35,503)( 36,502)( 37,487)( 38,489)( 39,488)( 40,491)
( 41,490)( 42,492)( 43,495)( 44,494)( 45,493)( 46,505)( 47,507)( 48,506)
( 49,509)( 50,508)( 51,510)( 52,513)( 53,512)( 54,511)( 55,550)( 56,552)
( 57,551)( 58,554)( 59,553)( 60,555)( 61,558)( 62,557)( 63,556)( 64,541)
( 65,543)( 66,542)( 67,545)( 68,544)( 69,546)( 70,549)( 71,548)( 72,547)
( 73,559)( 74,561)( 75,560)( 76,563)( 77,562)( 78,564)( 79,567)( 80,566)
( 81,565)( 82,604)( 83,606)( 84,605)( 85,608)( 86,607)( 87,609)( 88,612)
( 89,611)( 90,610)( 91,595)( 92,597)( 93,596)( 94,599)( 95,598)( 96,600)
( 97,603)( 98,602)( 99,601)(100,613)(101,615)(102,614)(103,617)(104,616)
(105,618)(106,621)(107,620)(108,619)(109,577)(110,579)(111,578)(112,581)
(113,580)(114,582)(115,585)(116,584)(117,583)(118,568)(119,570)(120,569)
(121,572)(122,571)(123,573)(124,576)(125,575)(126,574)(127,586)(128,588)
(129,587)(130,590)(131,589)(132,591)(133,594)(134,593)(135,592)(136,631)
(137,633)(138,632)(139,635)(140,634)(141,636)(142,639)(143,638)(144,637)
(145,622)(146,624)(147,623)(148,626)(149,625)(150,627)(151,630)(152,629)
(153,628)(154,640)(155,642)(156,641)(157,644)(158,643)(159,645)(160,648)
(161,647)(162,646)(163,442)(164,444)(165,443)(166,446)(167,445)(168,447)
(169,450)(170,449)(171,448)(172,433)(173,435)(174,434)(175,437)(176,436)
(177,438)(178,441)(179,440)(180,439)(181,451)(182,453)(183,452)(184,455)
(185,454)(186,456)(187,459)(188,458)(189,457)(190,415)(191,417)(192,416)
(193,419)(194,418)(195,420)(196,423)(197,422)(198,421)(199,406)(200,408)
(201,407)(202,410)(203,409)(204,411)(205,414)(206,413)(207,412)(208,424)
(209,426)(210,425)(211,428)(212,427)(213,429)(214,432)(215,431)(216,430)
(217,469)(218,471)(219,470)(220,473)(221,472)(222,474)(223,477)(224,476)
(225,475)(226,460)(227,462)(228,461)(229,464)(230,463)(231,465)(232,468)
(233,467)(234,466)(235,478)(236,480)(237,479)(238,482)(239,481)(240,483)
(241,486)(242,485)(243,484)(244,361)(245,363)(246,362)(247,365)(248,364)
(249,366)(250,369)(251,368)(252,367)(253,352)(254,354)(255,353)(256,356)
(257,355)(258,357)(259,360)(260,359)(261,358)(262,370)(263,372)(264,371)
(265,374)(266,373)(267,375)(268,378)(269,377)(270,376)(271,334)(272,336)
(273,335)(274,338)(275,337)(276,339)(277,342)(278,341)(279,340)(280,325)
(281,327)(282,326)(283,329)(284,328)(285,330)(286,333)(287,332)(288,331)
(289,343)(290,345)(291,344)(292,347)(293,346)(294,348)(295,351)(296,350)
(297,349)(298,388)(299,390)(300,389)(301,392)(302,391)(303,393)(304,396)
(305,395)(306,394)(307,379)(308,381)(309,380)(310,383)(311,382)(312,384)
(313,387)(314,386)(315,385)(316,397)(317,399)(318,398)(319,401)(320,400)
(321,402)(322,405)(323,404)(324,403);;
s2 := ( 1,166)( 2,168)( 3,167)( 4,163)( 5,165)( 6,164)( 7,169)( 8,171)
( 9,170)( 10,184)( 11,186)( 12,185)( 13,181)( 14,183)( 15,182)( 16,187)
( 17,189)( 18,188)( 19,175)( 20,177)( 21,176)( 22,172)( 23,174)( 24,173)
( 25,178)( 26,180)( 27,179)( 28,193)( 29,195)( 30,194)( 31,190)( 32,192)
( 33,191)( 34,196)( 35,198)( 36,197)( 37,211)( 38,213)( 39,212)( 40,208)
( 41,210)( 42,209)( 43,214)( 44,216)( 45,215)( 46,202)( 47,204)( 48,203)
( 49,199)( 50,201)( 51,200)( 52,205)( 53,207)( 54,206)( 55,220)( 56,222)
( 57,221)( 58,217)( 59,219)( 60,218)( 61,223)( 62,225)( 63,224)( 64,238)
( 65,240)( 66,239)( 67,235)( 68,237)( 69,236)( 70,241)( 71,243)( 72,242)
( 73,229)( 74,231)( 75,230)( 76,226)( 77,228)( 78,227)( 79,232)( 80,234)
( 81,233)( 82,247)( 83,249)( 84,248)( 85,244)( 86,246)( 87,245)( 88,250)
( 89,252)( 90,251)( 91,265)( 92,267)( 93,266)( 94,262)( 95,264)( 96,263)
( 97,268)( 98,270)( 99,269)(100,256)(101,258)(102,257)(103,253)(104,255)
(105,254)(106,259)(107,261)(108,260)(109,274)(110,276)(111,275)(112,271)
(113,273)(114,272)(115,277)(116,279)(117,278)(118,292)(119,294)(120,293)
(121,289)(122,291)(123,290)(124,295)(125,297)(126,296)(127,283)(128,285)
(129,284)(130,280)(131,282)(132,281)(133,286)(134,288)(135,287)(136,301)
(137,303)(138,302)(139,298)(140,300)(141,299)(142,304)(143,306)(144,305)
(145,319)(146,321)(147,320)(148,316)(149,318)(150,317)(151,322)(152,324)
(153,323)(154,310)(155,312)(156,311)(157,307)(158,309)(159,308)(160,313)
(161,315)(162,314)(325,571)(326,573)(327,572)(328,568)(329,570)(330,569)
(331,574)(332,576)(333,575)(334,589)(335,591)(336,590)(337,586)(338,588)
(339,587)(340,592)(341,594)(342,593)(343,580)(344,582)(345,581)(346,577)
(347,579)(348,578)(349,583)(350,585)(351,584)(352,598)(353,600)(354,599)
(355,595)(356,597)(357,596)(358,601)(359,603)(360,602)(361,616)(362,618)
(363,617)(364,613)(365,615)(366,614)(367,619)(368,621)(369,620)(370,607)
(371,609)(372,608)(373,604)(374,606)(375,605)(376,610)(377,612)(378,611)
(379,625)(380,627)(381,626)(382,622)(383,624)(384,623)(385,628)(386,630)
(387,629)(388,643)(389,645)(390,644)(391,640)(392,642)(393,641)(394,646)
(395,648)(396,647)(397,634)(398,636)(399,635)(400,631)(401,633)(402,632)
(403,637)(404,639)(405,638)(406,490)(407,492)(408,491)(409,487)(410,489)
(411,488)(412,493)(413,495)(414,494)(415,508)(416,510)(417,509)(418,505)
(419,507)(420,506)(421,511)(422,513)(423,512)(424,499)(425,501)(426,500)
(427,496)(428,498)(429,497)(430,502)(431,504)(432,503)(433,517)(434,519)
(435,518)(436,514)(437,516)(438,515)(439,520)(440,522)(441,521)(442,535)
(443,537)(444,536)(445,532)(446,534)(447,533)(448,538)(449,540)(450,539)
(451,526)(452,528)(453,527)(454,523)(455,525)(456,524)(457,529)(458,531)
(459,530)(460,544)(461,546)(462,545)(463,541)(464,543)(465,542)(466,547)
(467,549)(468,548)(469,562)(470,564)(471,563)(472,559)(473,561)(474,560)
(475,565)(476,567)(477,566)(478,553)(479,555)(480,554)(481,550)(482,552)
(483,551)(484,556)(485,558)(486,557);;
s3 := ( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)( 15, 17)
( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)( 33, 35)
( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)( 51, 53)
( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)( 69, 71)
( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)( 87, 89)
( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)(105,107)
(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)(123,125)
(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)(141,143)
(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)(159,161)
(163,244)(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)(170,249)
(171,248)(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)(178,256)
(179,258)(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)(186,269)
(187,265)(188,267)(189,266)(190,271)(191,273)(192,272)(193,277)(194,279)
(195,278)(196,274)(197,276)(198,275)(199,280)(200,282)(201,281)(202,286)
(203,288)(204,287)(205,283)(206,285)(207,284)(208,289)(209,291)(210,290)
(211,295)(212,297)(213,296)(214,292)(215,294)(216,293)(217,298)(218,300)
(219,299)(220,304)(221,306)(222,305)(223,301)(224,303)(225,302)(226,307)
(227,309)(228,308)(229,313)(230,315)(231,314)(232,310)(233,312)(234,311)
(235,316)(236,318)(237,317)(238,322)(239,324)(240,323)(241,319)(242,321)
(243,320)(325,406)(326,408)(327,407)(328,412)(329,414)(330,413)(331,409)
(332,411)(333,410)(334,415)(335,417)(336,416)(337,421)(338,423)(339,422)
(340,418)(341,420)(342,419)(343,424)(344,426)(345,425)(346,430)(347,432)
(348,431)(349,427)(350,429)(351,428)(352,433)(353,435)(354,434)(355,439)
(356,441)(357,440)(358,436)(359,438)(360,437)(361,442)(362,444)(363,443)
(364,448)(365,450)(366,449)(367,445)(368,447)(369,446)(370,451)(371,453)
(372,452)(373,457)(374,459)(375,458)(376,454)(377,456)(378,455)(379,460)
(380,462)(381,461)(382,466)(383,468)(384,467)(385,463)(386,465)(387,464)
(388,469)(389,471)(390,470)(391,475)(392,477)(393,476)(394,472)(395,474)
(396,473)(397,478)(398,480)(399,479)(400,484)(401,486)(402,485)(403,481)
(404,483)(405,482)(488,489)(490,493)(491,495)(492,494)(497,498)(499,502)
(500,504)(501,503)(506,507)(508,511)(509,513)(510,512)(515,516)(517,520)
(518,522)(519,521)(524,525)(526,529)(527,531)(528,530)(533,534)(535,538)
(536,540)(537,539)(542,543)(544,547)(545,549)(546,548)(551,552)(553,556)
(554,558)(555,557)(560,561)(562,565)(563,567)(564,566)(569,570)(571,574)
(572,576)(573,575)(578,579)(580,583)(581,585)(582,584)(587,588)(589,592)
(590,594)(591,593)(596,597)(598,601)(599,603)(600,602)(605,606)(607,610)
(608,612)(609,611)(614,615)(616,619)(617,621)(618,620)(623,624)(625,628)
(626,630)(627,629)(632,633)(634,637)(635,639)(636,638)(641,642)(643,646)
(644,648)(645,647);;
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,
s2*s0*s1*s2*s1*s2*s0*s1*s2*s1, s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(648)!( 1,487)( 2,489)( 3,488)( 4,490)( 5,492)( 6,491)( 7,493)
( 8,495)( 9,494)( 10,505)( 11,507)( 12,506)( 13,508)( 14,510)( 15,509)
( 16,511)( 17,513)( 18,512)( 19,496)( 20,498)( 21,497)( 22,499)( 23,501)
( 24,500)( 25,502)( 26,504)( 27,503)( 28,541)( 29,543)( 30,542)( 31,544)
( 32,546)( 33,545)( 34,547)( 35,549)( 36,548)( 37,559)( 38,561)( 39,560)
( 40,562)( 41,564)( 42,563)( 43,565)( 44,567)( 45,566)( 46,550)( 47,552)
( 48,551)( 49,553)( 50,555)( 51,554)( 52,556)( 53,558)( 54,557)( 55,514)
( 56,516)( 57,515)( 58,517)( 59,519)( 60,518)( 61,520)( 62,522)( 63,521)
( 64,532)( 65,534)( 66,533)( 67,535)( 68,537)( 69,536)( 70,538)( 71,540)
( 72,539)( 73,523)( 74,525)( 75,524)( 76,526)( 77,528)( 78,527)( 79,529)
( 80,531)( 81,530)( 82,568)( 83,570)( 84,569)( 85,571)( 86,573)( 87,572)
( 88,574)( 89,576)( 90,575)( 91,586)( 92,588)( 93,587)( 94,589)( 95,591)
( 96,590)( 97,592)( 98,594)( 99,593)(100,577)(101,579)(102,578)(103,580)
(104,582)(105,581)(106,583)(107,585)(108,584)(109,622)(110,624)(111,623)
(112,625)(113,627)(114,626)(115,628)(116,630)(117,629)(118,640)(119,642)
(120,641)(121,643)(122,645)(123,644)(124,646)(125,648)(126,647)(127,631)
(128,633)(129,632)(130,634)(131,636)(132,635)(133,637)(134,639)(135,638)
(136,595)(137,597)(138,596)(139,598)(140,600)(141,599)(142,601)(143,603)
(144,602)(145,613)(146,615)(147,614)(148,616)(149,618)(150,617)(151,619)
(152,621)(153,620)(154,604)(155,606)(156,605)(157,607)(158,609)(159,608)
(160,610)(161,612)(162,611)(163,406)(164,408)(165,407)(166,409)(167,411)
(168,410)(169,412)(170,414)(171,413)(172,424)(173,426)(174,425)(175,427)
(176,429)(177,428)(178,430)(179,432)(180,431)(181,415)(182,417)(183,416)
(184,418)(185,420)(186,419)(187,421)(188,423)(189,422)(190,460)(191,462)
(192,461)(193,463)(194,465)(195,464)(196,466)(197,468)(198,467)(199,478)
(200,480)(201,479)(202,481)(203,483)(204,482)(205,484)(206,486)(207,485)
(208,469)(209,471)(210,470)(211,472)(212,474)(213,473)(214,475)(215,477)
(216,476)(217,433)(218,435)(219,434)(220,436)(221,438)(222,437)(223,439)
(224,441)(225,440)(226,451)(227,453)(228,452)(229,454)(230,456)(231,455)
(232,457)(233,459)(234,458)(235,442)(236,444)(237,443)(238,445)(239,447)
(240,446)(241,448)(242,450)(243,449)(244,325)(245,327)(246,326)(247,328)
(248,330)(249,329)(250,331)(251,333)(252,332)(253,343)(254,345)(255,344)
(256,346)(257,348)(258,347)(259,349)(260,351)(261,350)(262,334)(263,336)
(264,335)(265,337)(266,339)(267,338)(268,340)(269,342)(270,341)(271,379)
(272,381)(273,380)(274,382)(275,384)(276,383)(277,385)(278,387)(279,386)
(280,397)(281,399)(282,398)(283,400)(284,402)(285,401)(286,403)(287,405)
(288,404)(289,388)(290,390)(291,389)(292,391)(293,393)(294,392)(295,394)
(296,396)(297,395)(298,352)(299,354)(300,353)(301,355)(302,357)(303,356)
(304,358)(305,360)(306,359)(307,370)(308,372)(309,371)(310,373)(311,375)
(312,374)(313,376)(314,378)(315,377)(316,361)(317,363)(318,362)(319,364)
(320,366)(321,365)(322,367)(323,369)(324,368);
s1 := Sym(648)!( 1,523)( 2,525)( 3,524)( 4,527)( 5,526)( 6,528)( 7,531)
( 8,530)( 9,529)( 10,514)( 11,516)( 12,515)( 13,518)( 14,517)( 15,519)
( 16,522)( 17,521)( 18,520)( 19,532)( 20,534)( 21,533)( 22,536)( 23,535)
( 24,537)( 25,540)( 26,539)( 27,538)( 28,496)( 29,498)( 30,497)( 31,500)
( 32,499)( 33,501)( 34,504)( 35,503)( 36,502)( 37,487)( 38,489)( 39,488)
( 40,491)( 41,490)( 42,492)( 43,495)( 44,494)( 45,493)( 46,505)( 47,507)
( 48,506)( 49,509)( 50,508)( 51,510)( 52,513)( 53,512)( 54,511)( 55,550)
( 56,552)( 57,551)( 58,554)( 59,553)( 60,555)( 61,558)( 62,557)( 63,556)
( 64,541)( 65,543)( 66,542)( 67,545)( 68,544)( 69,546)( 70,549)( 71,548)
( 72,547)( 73,559)( 74,561)( 75,560)( 76,563)( 77,562)( 78,564)( 79,567)
( 80,566)( 81,565)( 82,604)( 83,606)( 84,605)( 85,608)( 86,607)( 87,609)
( 88,612)( 89,611)( 90,610)( 91,595)( 92,597)( 93,596)( 94,599)( 95,598)
( 96,600)( 97,603)( 98,602)( 99,601)(100,613)(101,615)(102,614)(103,617)
(104,616)(105,618)(106,621)(107,620)(108,619)(109,577)(110,579)(111,578)
(112,581)(113,580)(114,582)(115,585)(116,584)(117,583)(118,568)(119,570)
(120,569)(121,572)(122,571)(123,573)(124,576)(125,575)(126,574)(127,586)
(128,588)(129,587)(130,590)(131,589)(132,591)(133,594)(134,593)(135,592)
(136,631)(137,633)(138,632)(139,635)(140,634)(141,636)(142,639)(143,638)
(144,637)(145,622)(146,624)(147,623)(148,626)(149,625)(150,627)(151,630)
(152,629)(153,628)(154,640)(155,642)(156,641)(157,644)(158,643)(159,645)
(160,648)(161,647)(162,646)(163,442)(164,444)(165,443)(166,446)(167,445)
(168,447)(169,450)(170,449)(171,448)(172,433)(173,435)(174,434)(175,437)
(176,436)(177,438)(178,441)(179,440)(180,439)(181,451)(182,453)(183,452)
(184,455)(185,454)(186,456)(187,459)(188,458)(189,457)(190,415)(191,417)
(192,416)(193,419)(194,418)(195,420)(196,423)(197,422)(198,421)(199,406)
(200,408)(201,407)(202,410)(203,409)(204,411)(205,414)(206,413)(207,412)
(208,424)(209,426)(210,425)(211,428)(212,427)(213,429)(214,432)(215,431)
(216,430)(217,469)(218,471)(219,470)(220,473)(221,472)(222,474)(223,477)
(224,476)(225,475)(226,460)(227,462)(228,461)(229,464)(230,463)(231,465)
(232,468)(233,467)(234,466)(235,478)(236,480)(237,479)(238,482)(239,481)
(240,483)(241,486)(242,485)(243,484)(244,361)(245,363)(246,362)(247,365)
(248,364)(249,366)(250,369)(251,368)(252,367)(253,352)(254,354)(255,353)
(256,356)(257,355)(258,357)(259,360)(260,359)(261,358)(262,370)(263,372)
(264,371)(265,374)(266,373)(267,375)(268,378)(269,377)(270,376)(271,334)
(272,336)(273,335)(274,338)(275,337)(276,339)(277,342)(278,341)(279,340)
(280,325)(281,327)(282,326)(283,329)(284,328)(285,330)(286,333)(287,332)
(288,331)(289,343)(290,345)(291,344)(292,347)(293,346)(294,348)(295,351)
(296,350)(297,349)(298,388)(299,390)(300,389)(301,392)(302,391)(303,393)
(304,396)(305,395)(306,394)(307,379)(308,381)(309,380)(310,383)(311,382)
(312,384)(313,387)(314,386)(315,385)(316,397)(317,399)(318,398)(319,401)
(320,400)(321,402)(322,405)(323,404)(324,403);
s2 := Sym(648)!( 1,166)( 2,168)( 3,167)( 4,163)( 5,165)( 6,164)( 7,169)
( 8,171)( 9,170)( 10,184)( 11,186)( 12,185)( 13,181)( 14,183)( 15,182)
( 16,187)( 17,189)( 18,188)( 19,175)( 20,177)( 21,176)( 22,172)( 23,174)
( 24,173)( 25,178)( 26,180)( 27,179)( 28,193)( 29,195)( 30,194)( 31,190)
( 32,192)( 33,191)( 34,196)( 35,198)( 36,197)( 37,211)( 38,213)( 39,212)
( 40,208)( 41,210)( 42,209)( 43,214)( 44,216)( 45,215)( 46,202)( 47,204)
( 48,203)( 49,199)( 50,201)( 51,200)( 52,205)( 53,207)( 54,206)( 55,220)
( 56,222)( 57,221)( 58,217)( 59,219)( 60,218)( 61,223)( 62,225)( 63,224)
( 64,238)( 65,240)( 66,239)( 67,235)( 68,237)( 69,236)( 70,241)( 71,243)
( 72,242)( 73,229)( 74,231)( 75,230)( 76,226)( 77,228)( 78,227)( 79,232)
( 80,234)( 81,233)( 82,247)( 83,249)( 84,248)( 85,244)( 86,246)( 87,245)
( 88,250)( 89,252)( 90,251)( 91,265)( 92,267)( 93,266)( 94,262)( 95,264)
( 96,263)( 97,268)( 98,270)( 99,269)(100,256)(101,258)(102,257)(103,253)
(104,255)(105,254)(106,259)(107,261)(108,260)(109,274)(110,276)(111,275)
(112,271)(113,273)(114,272)(115,277)(116,279)(117,278)(118,292)(119,294)
(120,293)(121,289)(122,291)(123,290)(124,295)(125,297)(126,296)(127,283)
(128,285)(129,284)(130,280)(131,282)(132,281)(133,286)(134,288)(135,287)
(136,301)(137,303)(138,302)(139,298)(140,300)(141,299)(142,304)(143,306)
(144,305)(145,319)(146,321)(147,320)(148,316)(149,318)(150,317)(151,322)
(152,324)(153,323)(154,310)(155,312)(156,311)(157,307)(158,309)(159,308)
(160,313)(161,315)(162,314)(325,571)(326,573)(327,572)(328,568)(329,570)
(330,569)(331,574)(332,576)(333,575)(334,589)(335,591)(336,590)(337,586)
(338,588)(339,587)(340,592)(341,594)(342,593)(343,580)(344,582)(345,581)
(346,577)(347,579)(348,578)(349,583)(350,585)(351,584)(352,598)(353,600)
(354,599)(355,595)(356,597)(357,596)(358,601)(359,603)(360,602)(361,616)
(362,618)(363,617)(364,613)(365,615)(366,614)(367,619)(368,621)(369,620)
(370,607)(371,609)(372,608)(373,604)(374,606)(375,605)(376,610)(377,612)
(378,611)(379,625)(380,627)(381,626)(382,622)(383,624)(384,623)(385,628)
(386,630)(387,629)(388,643)(389,645)(390,644)(391,640)(392,642)(393,641)
(394,646)(395,648)(396,647)(397,634)(398,636)(399,635)(400,631)(401,633)
(402,632)(403,637)(404,639)(405,638)(406,490)(407,492)(408,491)(409,487)
(410,489)(411,488)(412,493)(413,495)(414,494)(415,508)(416,510)(417,509)
(418,505)(419,507)(420,506)(421,511)(422,513)(423,512)(424,499)(425,501)
(426,500)(427,496)(428,498)(429,497)(430,502)(431,504)(432,503)(433,517)
(434,519)(435,518)(436,514)(437,516)(438,515)(439,520)(440,522)(441,521)
(442,535)(443,537)(444,536)(445,532)(446,534)(447,533)(448,538)(449,540)
(450,539)(451,526)(452,528)(453,527)(454,523)(455,525)(456,524)(457,529)
(458,531)(459,530)(460,544)(461,546)(462,545)(463,541)(464,543)(465,542)
(466,547)(467,549)(468,548)(469,562)(470,564)(471,563)(472,559)(473,561)
(474,560)(475,565)(476,567)(477,566)(478,553)(479,555)(480,554)(481,550)
(482,552)(483,551)(484,556)(485,558)(486,557);
s3 := Sym(648)!( 2, 3)( 4, 7)( 5, 9)( 6, 8)( 11, 12)( 13, 16)( 14, 18)
( 15, 17)( 20, 21)( 22, 25)( 23, 27)( 24, 26)( 29, 30)( 31, 34)( 32, 36)
( 33, 35)( 38, 39)( 40, 43)( 41, 45)( 42, 44)( 47, 48)( 49, 52)( 50, 54)
( 51, 53)( 56, 57)( 58, 61)( 59, 63)( 60, 62)( 65, 66)( 67, 70)( 68, 72)
( 69, 71)( 74, 75)( 76, 79)( 77, 81)( 78, 80)( 83, 84)( 85, 88)( 86, 90)
( 87, 89)( 92, 93)( 94, 97)( 95, 99)( 96, 98)(101,102)(103,106)(104,108)
(105,107)(110,111)(112,115)(113,117)(114,116)(119,120)(121,124)(122,126)
(123,125)(128,129)(130,133)(131,135)(132,134)(137,138)(139,142)(140,144)
(141,143)(146,147)(148,151)(149,153)(150,152)(155,156)(157,160)(158,162)
(159,161)(163,244)(164,246)(165,245)(166,250)(167,252)(168,251)(169,247)
(170,249)(171,248)(172,253)(173,255)(174,254)(175,259)(176,261)(177,260)
(178,256)(179,258)(180,257)(181,262)(182,264)(183,263)(184,268)(185,270)
(186,269)(187,265)(188,267)(189,266)(190,271)(191,273)(192,272)(193,277)
(194,279)(195,278)(196,274)(197,276)(198,275)(199,280)(200,282)(201,281)
(202,286)(203,288)(204,287)(205,283)(206,285)(207,284)(208,289)(209,291)
(210,290)(211,295)(212,297)(213,296)(214,292)(215,294)(216,293)(217,298)
(218,300)(219,299)(220,304)(221,306)(222,305)(223,301)(224,303)(225,302)
(226,307)(227,309)(228,308)(229,313)(230,315)(231,314)(232,310)(233,312)
(234,311)(235,316)(236,318)(237,317)(238,322)(239,324)(240,323)(241,319)
(242,321)(243,320)(325,406)(326,408)(327,407)(328,412)(329,414)(330,413)
(331,409)(332,411)(333,410)(334,415)(335,417)(336,416)(337,421)(338,423)
(339,422)(340,418)(341,420)(342,419)(343,424)(344,426)(345,425)(346,430)
(347,432)(348,431)(349,427)(350,429)(351,428)(352,433)(353,435)(354,434)
(355,439)(356,441)(357,440)(358,436)(359,438)(360,437)(361,442)(362,444)
(363,443)(364,448)(365,450)(366,449)(367,445)(368,447)(369,446)(370,451)
(371,453)(372,452)(373,457)(374,459)(375,458)(376,454)(377,456)(378,455)
(379,460)(380,462)(381,461)(382,466)(383,468)(384,467)(385,463)(386,465)
(387,464)(388,469)(389,471)(390,470)(391,475)(392,477)(393,476)(394,472)
(395,474)(396,473)(397,478)(398,480)(399,479)(400,484)(401,486)(402,485)
(403,481)(404,483)(405,482)(488,489)(490,493)(491,495)(492,494)(497,498)
(499,502)(500,504)(501,503)(506,507)(508,511)(509,513)(510,512)(515,516)
(517,520)(518,522)(519,521)(524,525)(526,529)(527,531)(528,530)(533,534)
(535,538)(536,540)(537,539)(542,543)(544,547)(545,549)(546,548)(551,552)
(553,556)(554,558)(555,557)(560,561)(562,565)(563,567)(564,566)(569,570)
(571,574)(572,576)(573,575)(578,579)(580,583)(581,585)(582,584)(587,588)
(589,592)(590,594)(591,593)(596,597)(598,601)(599,603)(600,602)(605,606)
(607,610)(608,612)(609,611)(614,615)(616,619)(617,621)(618,620)(623,624)
(625,628)(626,630)(627,629)(632,633)(634,637)(635,639)(636,638)(641,642)
(643,646)(644,648)(645,647);
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*s0*s1*s0*s1, s2*s0*s1*s2*s1*s2*s0*s1*s2*s1,
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3 >;
References : None.
to this polytope