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# Polytope of Type {6,4,4,6}

Atlas Canonical Name : {6,4,4,6}*1152
Also Known As : {{6,4|2},{4,4|2},{4,6|2}}. if this polytope has another name.
Group : SmallGroup(1152,134261)
Rank : 5
Schlafli Type : {6,4,4,6}
Number of vertices, edges, etc : 6, 12, 8, 12, 6
Order of s0s1s2s3s4 : 12
Order of s0s1s2s3s4s3s2s1 : 2
Special Properties :
Universal
Orientable
Flat
Self-Dual
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 : {6,2,4,6}*576a, {6,4,2,6}*576a
3-fold quotients : {2,4,4,6}*384, {6,4,4,2}*384
4-fold quotients : {3,2,4,6}*288a, {6,4,2,3}*288a, {6,2,2,6}*288
6-fold quotients : {2,2,4,6}*192a, {2,4,2,6}*192, {6,2,4,2}*192, {6,4,2,2}*192a
8-fold quotients : {3,2,2,6}*144, {6,2,2,3}*144
9-fold quotients : {2,4,4,2}*128
12-fold quotients : {2,4,2,3}*96, {3,2,4,2}*96, {2,2,2,6}*96, {6,2,2,2}*96
16-fold quotients : {3,2,2,3}*72
18-fold quotients : {2,2,4,2}*64, {2,4,2,2}*64
24-fold quotients : {2,2,2,3}*48, {3,2,2,2}*48
36-fold quotients : {2,2,2,2}*32
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (  1, 37)(  2, 42)(  3, 44)(  4, 40)(  5, 45)(  6, 38)(  7, 43)(  8, 39)
(  9, 41)( 10, 46)( 11, 51)( 12, 53)( 13, 49)( 14, 54)( 15, 47)( 16, 52)
( 17, 48)( 18, 50)( 19, 55)( 20, 60)( 21, 62)( 22, 58)( 23, 63)( 24, 56)
( 25, 61)( 26, 57)( 27, 59)( 28, 64)( 29, 69)( 30, 71)( 31, 67)( 32, 72)
( 33, 65)( 34, 70)( 35, 66)( 36, 68)( 73,109)( 74,114)( 75,116)( 76,112)
( 77,117)( 78,110)( 79,115)( 80,111)( 81,113)( 82,118)( 83,123)( 84,125)
( 85,121)( 86,126)( 87,119)( 88,124)( 89,120)( 90,122)( 91,127)( 92,132)
( 93,134)( 94,130)( 95,135)( 96,128)( 97,133)( 98,129)( 99,131)(100,136)
(101,141)(102,143)(103,139)(104,144)(105,137)(106,142)(107,138)(108,140)
(145,181)(146,186)(147,188)(148,184)(149,189)(150,182)(151,187)(152,183)
(153,185)(154,190)(155,195)(156,197)(157,193)(158,198)(159,191)(160,196)
(161,192)(162,194)(163,199)(164,204)(165,206)(166,202)(167,207)(168,200)
(169,205)(170,201)(171,203)(172,208)(173,213)(174,215)(175,211)(176,216)
(177,209)(178,214)(179,210)(180,212)(217,253)(218,258)(219,260)(220,256)
(221,261)(222,254)(223,259)(224,255)(225,257)(226,262)(227,267)(228,269)
(229,265)(230,270)(231,263)(232,268)(233,264)(234,266)(235,271)(236,276)
(237,278)(238,274)(239,279)(240,272)(241,277)(242,273)(243,275)(244,280)
(245,285)(246,287)(247,283)(248,288)(249,281)(250,286)(251,282)(252,284)
(289,325)(290,330)(291,332)(292,328)(293,333)(294,326)(295,331)(296,327)
(297,329)(298,334)(299,339)(300,341)(301,337)(302,342)(303,335)(304,340)
(305,336)(306,338)(307,343)(308,348)(309,350)(310,346)(311,351)(312,344)
(313,349)(314,345)(315,347)(316,352)(317,357)(318,359)(319,355)(320,360)
(321,353)(322,358)(323,354)(324,356)(361,397)(362,402)(363,404)(364,400)
(365,405)(366,398)(367,403)(368,399)(369,401)(370,406)(371,411)(372,413)
(373,409)(374,414)(375,407)(376,412)(377,408)(378,410)(379,415)(380,420)
(381,422)(382,418)(383,423)(384,416)(385,421)(386,417)(387,419)(388,424)
(389,429)(390,431)(391,427)(392,432)(393,425)(394,430)(395,426)(396,428)
(433,469)(434,474)(435,476)(436,472)(437,477)(438,470)(439,475)(440,471)
(441,473)(442,478)(443,483)(444,485)(445,481)(446,486)(447,479)(448,484)
(449,480)(450,482)(451,487)(452,492)(453,494)(454,490)(455,495)(456,488)
(457,493)(458,489)(459,491)(460,496)(461,501)(462,503)(463,499)(464,504)
(465,497)(466,502)(467,498)(468,500)(505,541)(506,546)(507,548)(508,544)
(509,549)(510,542)(511,547)(512,543)(513,545)(514,550)(515,555)(516,557)
(517,553)(518,558)(519,551)(520,556)(521,552)(522,554)(523,559)(524,564)
(525,566)(526,562)(527,567)(528,560)(529,565)(530,561)(531,563)(532,568)
(533,573)(534,575)(535,571)(536,576)(537,569)(538,574)(539,570)(540,572);;
s1 := (  1,329)(  2,331)(  3,327)(  4,332)(  5,325)(  6,330)(  7,326)(  8,328)
(  9,333)( 10,338)( 11,340)( 12,336)( 13,341)( 14,334)( 15,339)( 16,335)
( 17,337)( 18,342)( 19,347)( 20,349)( 21,345)( 22,350)( 23,343)( 24,348)
( 25,344)( 26,346)( 27,351)( 28,356)( 29,358)( 30,354)( 31,359)( 32,352)
( 33,357)( 34,353)( 35,355)( 36,360)( 37,293)( 38,295)( 39,291)( 40,296)
( 41,289)( 42,294)( 43,290)( 44,292)( 45,297)( 46,302)( 47,304)( 48,300)
( 49,305)( 50,298)( 51,303)( 52,299)( 53,301)( 54,306)( 55,311)( 56,313)
( 57,309)( 58,314)( 59,307)( 60,312)( 61,308)( 62,310)( 63,315)( 64,320)
( 65,322)( 66,318)( 67,323)( 68,316)( 69,321)( 70,317)( 71,319)( 72,324)
( 73,401)( 74,403)( 75,399)( 76,404)( 77,397)( 78,402)( 79,398)( 80,400)
( 81,405)( 82,410)( 83,412)( 84,408)( 85,413)( 86,406)( 87,411)( 88,407)
( 89,409)( 90,414)( 91,419)( 92,421)( 93,417)( 94,422)( 95,415)( 96,420)
( 97,416)( 98,418)( 99,423)(100,428)(101,430)(102,426)(103,431)(104,424)
(105,429)(106,425)(107,427)(108,432)(109,365)(110,367)(111,363)(112,368)
(113,361)(114,366)(115,362)(116,364)(117,369)(118,374)(119,376)(120,372)
(121,377)(122,370)(123,375)(124,371)(125,373)(126,378)(127,383)(128,385)
(129,381)(130,386)(131,379)(132,384)(133,380)(134,382)(135,387)(136,392)
(137,394)(138,390)(139,395)(140,388)(141,393)(142,389)(143,391)(144,396)
(145,473)(146,475)(147,471)(148,476)(149,469)(150,474)(151,470)(152,472)
(153,477)(154,482)(155,484)(156,480)(157,485)(158,478)(159,483)(160,479)
(161,481)(162,486)(163,491)(164,493)(165,489)(166,494)(167,487)(168,492)
(169,488)(170,490)(171,495)(172,500)(173,502)(174,498)(175,503)(176,496)
(177,501)(178,497)(179,499)(180,504)(181,437)(182,439)(183,435)(184,440)
(185,433)(186,438)(187,434)(188,436)(189,441)(190,446)(191,448)(192,444)
(193,449)(194,442)(195,447)(196,443)(197,445)(198,450)(199,455)(200,457)
(201,453)(202,458)(203,451)(204,456)(205,452)(206,454)(207,459)(208,464)
(209,466)(210,462)(211,467)(212,460)(213,465)(214,461)(215,463)(216,468)
(217,545)(218,547)(219,543)(220,548)(221,541)(222,546)(223,542)(224,544)
(225,549)(226,554)(227,556)(228,552)(229,557)(230,550)(231,555)(232,551)
(233,553)(234,558)(235,563)(236,565)(237,561)(238,566)(239,559)(240,564)
(241,560)(242,562)(243,567)(244,572)(245,574)(246,570)(247,575)(248,568)
(249,573)(250,569)(251,571)(252,576)(253,509)(254,511)(255,507)(256,512)
(257,505)(258,510)(259,506)(260,508)(261,513)(262,518)(263,520)(264,516)
(265,521)(266,514)(267,519)(268,515)(269,517)(270,522)(271,527)(272,529)
(273,525)(274,530)(275,523)(276,528)(277,524)(278,526)(279,531)(280,536)
(281,538)(282,534)(283,539)(284,532)(285,537)(286,533)(287,535)(288,540);;
s2 := (145,154)(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)(152,161)
(153,162)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)(169,178)
(170,179)(171,180)(181,190)(182,191)(183,192)(184,193)(185,194)(186,195)
(187,196)(188,197)(189,198)(199,208)(200,209)(201,210)(202,211)(203,212)
(204,213)(205,214)(206,215)(207,216)(217,226)(218,227)(219,228)(220,229)
(221,230)(222,231)(223,232)(224,233)(225,234)(235,244)(236,245)(237,246)
(238,247)(239,248)(240,249)(241,250)(242,251)(243,252)(253,262)(254,263)
(255,264)(256,265)(257,266)(258,267)(259,268)(260,269)(261,270)(271,280)
(272,281)(273,282)(274,283)(275,284)(276,285)(277,286)(278,287)(279,288)
(289,307)(290,308)(291,309)(292,310)(293,311)(294,312)(295,313)(296,314)
(297,315)(298,316)(299,317)(300,318)(301,319)(302,320)(303,321)(304,322)
(305,323)(306,324)(325,343)(326,344)(327,345)(328,346)(329,347)(330,348)
(331,349)(332,350)(333,351)(334,352)(335,353)(336,354)(337,355)(338,356)
(339,357)(340,358)(341,359)(342,360)(361,379)(362,380)(363,381)(364,382)
(365,383)(366,384)(367,385)(368,386)(369,387)(370,388)(371,389)(372,390)
(373,391)(374,392)(375,393)(376,394)(377,395)(378,396)(397,415)(398,416)
(399,417)(400,418)(401,419)(402,420)(403,421)(404,422)(405,423)(406,424)
(407,425)(408,426)(409,427)(410,428)(411,429)(412,430)(413,431)(414,432)
(433,460)(434,461)(435,462)(436,463)(437,464)(438,465)(439,466)(440,467)
(441,468)(442,451)(443,452)(444,453)(445,454)(446,455)(447,456)(448,457)
(449,458)(450,459)(469,496)(470,497)(471,498)(472,499)(473,500)(474,501)
(475,502)(476,503)(477,504)(478,487)(479,488)(480,489)(481,490)(482,491)
(483,492)(484,493)(485,494)(486,495)(505,532)(506,533)(507,534)(508,535)
(509,536)(510,537)(511,538)(512,539)(513,540)(514,523)(515,524)(516,525)
(517,526)(518,527)(519,528)(520,529)(521,530)(522,531)(541,568)(542,569)
(543,570)(544,571)(545,572)(546,573)(547,574)(548,575)(549,576)(550,559)
(551,560)(552,561)(553,562)(554,563)(555,564)(556,565)(557,566)(558,567);;
s3 := (  1,253)(  2,260)(  3,258)(  4,259)(  5,257)(  6,255)(  7,256)(  8,254)
(  9,261)( 10,262)( 11,269)( 12,267)( 13,268)( 14,266)( 15,264)( 16,265)
( 17,263)( 18,270)( 19,271)( 20,278)( 21,276)( 22,277)( 23,275)( 24,273)
( 25,274)( 26,272)( 27,279)( 28,280)( 29,287)( 30,285)( 31,286)( 32,284)
( 33,282)( 34,283)( 35,281)( 36,288)( 37,217)( 38,224)( 39,222)( 40,223)
( 41,221)( 42,219)( 43,220)( 44,218)( 45,225)( 46,226)( 47,233)( 48,231)
( 49,232)( 50,230)( 51,228)( 52,229)( 53,227)( 54,234)( 55,235)( 56,242)
( 57,240)( 58,241)( 59,239)( 60,237)( 61,238)( 62,236)( 63,243)( 64,244)
( 65,251)( 66,249)( 67,250)( 68,248)( 69,246)( 70,247)( 71,245)( 72,252)
( 73,181)( 74,188)( 75,186)( 76,187)( 77,185)( 78,183)( 79,184)( 80,182)
( 81,189)( 82,190)( 83,197)( 84,195)( 85,196)( 86,194)( 87,192)( 88,193)
( 89,191)( 90,198)( 91,199)( 92,206)( 93,204)( 94,205)( 95,203)( 96,201)
( 97,202)( 98,200)( 99,207)(100,208)(101,215)(102,213)(103,214)(104,212)
(105,210)(106,211)(107,209)(108,216)(109,145)(110,152)(111,150)(112,151)
(113,149)(114,147)(115,148)(116,146)(117,153)(118,154)(119,161)(120,159)
(121,160)(122,158)(123,156)(124,157)(125,155)(126,162)(127,163)(128,170)
(129,168)(130,169)(131,167)(132,165)(133,166)(134,164)(135,171)(136,172)
(137,179)(138,177)(139,178)(140,176)(141,174)(142,175)(143,173)(144,180)
(289,541)(290,548)(291,546)(292,547)(293,545)(294,543)(295,544)(296,542)
(297,549)(298,550)(299,557)(300,555)(301,556)(302,554)(303,552)(304,553)
(305,551)(306,558)(307,559)(308,566)(309,564)(310,565)(311,563)(312,561)
(313,562)(314,560)(315,567)(316,568)(317,575)(318,573)(319,574)(320,572)
(321,570)(322,571)(323,569)(324,576)(325,505)(326,512)(327,510)(328,511)
(329,509)(330,507)(331,508)(332,506)(333,513)(334,514)(335,521)(336,519)
(337,520)(338,518)(339,516)(340,517)(341,515)(342,522)(343,523)(344,530)
(345,528)(346,529)(347,527)(348,525)(349,526)(350,524)(351,531)(352,532)
(353,539)(354,537)(355,538)(356,536)(357,534)(358,535)(359,533)(360,540)
(361,469)(362,476)(363,474)(364,475)(365,473)(366,471)(367,472)(368,470)
(369,477)(370,478)(371,485)(372,483)(373,484)(374,482)(375,480)(376,481)
(377,479)(378,486)(379,487)(380,494)(381,492)(382,493)(383,491)(384,489)
(385,490)(386,488)(387,495)(388,496)(389,503)(390,501)(391,502)(392,500)
(393,498)(394,499)(395,497)(396,504)(397,433)(398,440)(399,438)(400,439)
(401,437)(402,435)(403,436)(404,434)(405,441)(406,442)(407,449)(408,447)
(409,448)(410,446)(411,444)(412,445)(413,443)(414,450)(415,451)(416,458)
(417,456)(418,457)(419,455)(420,453)(421,454)(422,452)(423,459)(424,460)
(425,467)(426,465)(427,466)(428,464)(429,462)(430,463)(431,461)(432,468);;
s4 := (  1,112)(  2,110)(  3,117)(  4,109)(  5,116)(  6,114)(  7,115)(  8,113)
(  9,111)( 10,121)( 11,119)( 12,126)( 13,118)( 14,125)( 15,123)( 16,124)
( 17,122)( 18,120)( 19,130)( 20,128)( 21,135)( 22,127)( 23,134)( 24,132)
( 25,133)( 26,131)( 27,129)( 28,139)( 29,137)( 30,144)( 31,136)( 32,143)
( 33,141)( 34,142)( 35,140)( 36,138)( 37, 76)( 38, 74)( 39, 81)( 40, 73)
( 41, 80)( 42, 78)( 43, 79)( 44, 77)( 45, 75)( 46, 85)( 47, 83)( 48, 90)
( 49, 82)( 50, 89)( 51, 87)( 52, 88)( 53, 86)( 54, 84)( 55, 94)( 56, 92)
( 57, 99)( 58, 91)( 59, 98)( 60, 96)( 61, 97)( 62, 95)( 63, 93)( 64,103)
( 65,101)( 66,108)( 67,100)( 68,107)( 69,105)( 70,106)( 71,104)( 72,102)
(145,256)(146,254)(147,261)(148,253)(149,260)(150,258)(151,259)(152,257)
(153,255)(154,265)(155,263)(156,270)(157,262)(158,269)(159,267)(160,268)
(161,266)(162,264)(163,274)(164,272)(165,279)(166,271)(167,278)(168,276)
(169,277)(170,275)(171,273)(172,283)(173,281)(174,288)(175,280)(176,287)
(177,285)(178,286)(179,284)(180,282)(181,220)(182,218)(183,225)(184,217)
(185,224)(186,222)(187,223)(188,221)(189,219)(190,229)(191,227)(192,234)
(193,226)(194,233)(195,231)(196,232)(197,230)(198,228)(199,238)(200,236)
(201,243)(202,235)(203,242)(204,240)(205,241)(206,239)(207,237)(208,247)
(209,245)(210,252)(211,244)(212,251)(213,249)(214,250)(215,248)(216,246)
(289,400)(290,398)(291,405)(292,397)(293,404)(294,402)(295,403)(296,401)
(297,399)(298,409)(299,407)(300,414)(301,406)(302,413)(303,411)(304,412)
(305,410)(306,408)(307,418)(308,416)(309,423)(310,415)(311,422)(312,420)
(313,421)(314,419)(315,417)(316,427)(317,425)(318,432)(319,424)(320,431)
(321,429)(322,430)(323,428)(324,426)(325,364)(326,362)(327,369)(328,361)
(329,368)(330,366)(331,367)(332,365)(333,363)(334,373)(335,371)(336,378)
(337,370)(338,377)(339,375)(340,376)(341,374)(342,372)(343,382)(344,380)
(345,387)(346,379)(347,386)(348,384)(349,385)(350,383)(351,381)(352,391)
(353,389)(354,396)(355,388)(356,395)(357,393)(358,394)(359,392)(360,390)
(433,544)(434,542)(435,549)(436,541)(437,548)(438,546)(439,547)(440,545)
(441,543)(442,553)(443,551)(444,558)(445,550)(446,557)(447,555)(448,556)
(449,554)(450,552)(451,562)(452,560)(453,567)(454,559)(455,566)(456,564)
(457,565)(458,563)(459,561)(460,571)(461,569)(462,576)(463,568)(464,575)
(465,573)(466,574)(467,572)(468,570)(469,508)(470,506)(471,513)(472,505)
(473,512)(474,510)(475,511)(476,509)(477,507)(478,517)(479,515)(480,522)
(481,514)(482,521)(483,519)(484,520)(485,518)(486,516)(487,526)(488,524)
(489,531)(490,523)(491,530)(492,528)(493,529)(494,527)(495,525)(496,535)
(497,533)(498,540)(499,532)(500,539)(501,537)(502,538)(503,536)(504,534);;
poly := Group([s0,s1,s2,s3,s4]);;

Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3","s4");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  s3 := F.4;;  s4 := F.5;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s4*s4, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s4*s0*s4,
s1*s4*s1*s4, s2*s4*s2*s4, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 ];;
poly := F / rels;;

Permutation Representation (Magma) :
s0 := Sym(576)!(  1, 37)(  2, 42)(  3, 44)(  4, 40)(  5, 45)(  6, 38)(  7, 43)
(  8, 39)(  9, 41)( 10, 46)( 11, 51)( 12, 53)( 13, 49)( 14, 54)( 15, 47)
( 16, 52)( 17, 48)( 18, 50)( 19, 55)( 20, 60)( 21, 62)( 22, 58)( 23, 63)
( 24, 56)( 25, 61)( 26, 57)( 27, 59)( 28, 64)( 29, 69)( 30, 71)( 31, 67)
( 32, 72)( 33, 65)( 34, 70)( 35, 66)( 36, 68)( 73,109)( 74,114)( 75,116)
( 76,112)( 77,117)( 78,110)( 79,115)( 80,111)( 81,113)( 82,118)( 83,123)
( 84,125)( 85,121)( 86,126)( 87,119)( 88,124)( 89,120)( 90,122)( 91,127)
( 92,132)( 93,134)( 94,130)( 95,135)( 96,128)( 97,133)( 98,129)( 99,131)
(100,136)(101,141)(102,143)(103,139)(104,144)(105,137)(106,142)(107,138)
(108,140)(145,181)(146,186)(147,188)(148,184)(149,189)(150,182)(151,187)
(152,183)(153,185)(154,190)(155,195)(156,197)(157,193)(158,198)(159,191)
(160,196)(161,192)(162,194)(163,199)(164,204)(165,206)(166,202)(167,207)
(168,200)(169,205)(170,201)(171,203)(172,208)(173,213)(174,215)(175,211)
(176,216)(177,209)(178,214)(179,210)(180,212)(217,253)(218,258)(219,260)
(220,256)(221,261)(222,254)(223,259)(224,255)(225,257)(226,262)(227,267)
(228,269)(229,265)(230,270)(231,263)(232,268)(233,264)(234,266)(235,271)
(236,276)(237,278)(238,274)(239,279)(240,272)(241,277)(242,273)(243,275)
(244,280)(245,285)(246,287)(247,283)(248,288)(249,281)(250,286)(251,282)
(252,284)(289,325)(290,330)(291,332)(292,328)(293,333)(294,326)(295,331)
(296,327)(297,329)(298,334)(299,339)(300,341)(301,337)(302,342)(303,335)
(304,340)(305,336)(306,338)(307,343)(308,348)(309,350)(310,346)(311,351)
(312,344)(313,349)(314,345)(315,347)(316,352)(317,357)(318,359)(319,355)
(320,360)(321,353)(322,358)(323,354)(324,356)(361,397)(362,402)(363,404)
(364,400)(365,405)(366,398)(367,403)(368,399)(369,401)(370,406)(371,411)
(372,413)(373,409)(374,414)(375,407)(376,412)(377,408)(378,410)(379,415)
(380,420)(381,422)(382,418)(383,423)(384,416)(385,421)(386,417)(387,419)
(388,424)(389,429)(390,431)(391,427)(392,432)(393,425)(394,430)(395,426)
(396,428)(433,469)(434,474)(435,476)(436,472)(437,477)(438,470)(439,475)
(440,471)(441,473)(442,478)(443,483)(444,485)(445,481)(446,486)(447,479)
(448,484)(449,480)(450,482)(451,487)(452,492)(453,494)(454,490)(455,495)
(456,488)(457,493)(458,489)(459,491)(460,496)(461,501)(462,503)(463,499)
(464,504)(465,497)(466,502)(467,498)(468,500)(505,541)(506,546)(507,548)
(508,544)(509,549)(510,542)(511,547)(512,543)(513,545)(514,550)(515,555)
(516,557)(517,553)(518,558)(519,551)(520,556)(521,552)(522,554)(523,559)
(524,564)(525,566)(526,562)(527,567)(528,560)(529,565)(530,561)(531,563)
(532,568)(533,573)(534,575)(535,571)(536,576)(537,569)(538,574)(539,570)
(540,572);
s1 := Sym(576)!(  1,329)(  2,331)(  3,327)(  4,332)(  5,325)(  6,330)(  7,326)
(  8,328)(  9,333)( 10,338)( 11,340)( 12,336)( 13,341)( 14,334)( 15,339)
( 16,335)( 17,337)( 18,342)( 19,347)( 20,349)( 21,345)( 22,350)( 23,343)
( 24,348)( 25,344)( 26,346)( 27,351)( 28,356)( 29,358)( 30,354)( 31,359)
( 32,352)( 33,357)( 34,353)( 35,355)( 36,360)( 37,293)( 38,295)( 39,291)
( 40,296)( 41,289)( 42,294)( 43,290)( 44,292)( 45,297)( 46,302)( 47,304)
( 48,300)( 49,305)( 50,298)( 51,303)( 52,299)( 53,301)( 54,306)( 55,311)
( 56,313)( 57,309)( 58,314)( 59,307)( 60,312)( 61,308)( 62,310)( 63,315)
( 64,320)( 65,322)( 66,318)( 67,323)( 68,316)( 69,321)( 70,317)( 71,319)
( 72,324)( 73,401)( 74,403)( 75,399)( 76,404)( 77,397)( 78,402)( 79,398)
( 80,400)( 81,405)( 82,410)( 83,412)( 84,408)( 85,413)( 86,406)( 87,411)
( 88,407)( 89,409)( 90,414)( 91,419)( 92,421)( 93,417)( 94,422)( 95,415)
( 96,420)( 97,416)( 98,418)( 99,423)(100,428)(101,430)(102,426)(103,431)
(104,424)(105,429)(106,425)(107,427)(108,432)(109,365)(110,367)(111,363)
(112,368)(113,361)(114,366)(115,362)(116,364)(117,369)(118,374)(119,376)
(120,372)(121,377)(122,370)(123,375)(124,371)(125,373)(126,378)(127,383)
(128,385)(129,381)(130,386)(131,379)(132,384)(133,380)(134,382)(135,387)
(136,392)(137,394)(138,390)(139,395)(140,388)(141,393)(142,389)(143,391)
(144,396)(145,473)(146,475)(147,471)(148,476)(149,469)(150,474)(151,470)
(152,472)(153,477)(154,482)(155,484)(156,480)(157,485)(158,478)(159,483)
(160,479)(161,481)(162,486)(163,491)(164,493)(165,489)(166,494)(167,487)
(168,492)(169,488)(170,490)(171,495)(172,500)(173,502)(174,498)(175,503)
(176,496)(177,501)(178,497)(179,499)(180,504)(181,437)(182,439)(183,435)
(184,440)(185,433)(186,438)(187,434)(188,436)(189,441)(190,446)(191,448)
(192,444)(193,449)(194,442)(195,447)(196,443)(197,445)(198,450)(199,455)
(200,457)(201,453)(202,458)(203,451)(204,456)(205,452)(206,454)(207,459)
(208,464)(209,466)(210,462)(211,467)(212,460)(213,465)(214,461)(215,463)
(216,468)(217,545)(218,547)(219,543)(220,548)(221,541)(222,546)(223,542)
(224,544)(225,549)(226,554)(227,556)(228,552)(229,557)(230,550)(231,555)
(232,551)(233,553)(234,558)(235,563)(236,565)(237,561)(238,566)(239,559)
(240,564)(241,560)(242,562)(243,567)(244,572)(245,574)(246,570)(247,575)
(248,568)(249,573)(250,569)(251,571)(252,576)(253,509)(254,511)(255,507)
(256,512)(257,505)(258,510)(259,506)(260,508)(261,513)(262,518)(263,520)
(264,516)(265,521)(266,514)(267,519)(268,515)(269,517)(270,522)(271,527)
(272,529)(273,525)(274,530)(275,523)(276,528)(277,524)(278,526)(279,531)
(280,536)(281,538)(282,534)(283,539)(284,532)(285,537)(286,533)(287,535)
(288,540);
s2 := Sym(576)!(145,154)(146,155)(147,156)(148,157)(149,158)(150,159)(151,160)
(152,161)(153,162)(163,172)(164,173)(165,174)(166,175)(167,176)(168,177)
(169,178)(170,179)(171,180)(181,190)(182,191)(183,192)(184,193)(185,194)
(186,195)(187,196)(188,197)(189,198)(199,208)(200,209)(201,210)(202,211)
(203,212)(204,213)(205,214)(206,215)(207,216)(217,226)(218,227)(219,228)
(220,229)(221,230)(222,231)(223,232)(224,233)(225,234)(235,244)(236,245)
(237,246)(238,247)(239,248)(240,249)(241,250)(242,251)(243,252)(253,262)
(254,263)(255,264)(256,265)(257,266)(258,267)(259,268)(260,269)(261,270)
(271,280)(272,281)(273,282)(274,283)(275,284)(276,285)(277,286)(278,287)
(279,288)(289,307)(290,308)(291,309)(292,310)(293,311)(294,312)(295,313)
(296,314)(297,315)(298,316)(299,317)(300,318)(301,319)(302,320)(303,321)
(304,322)(305,323)(306,324)(325,343)(326,344)(327,345)(328,346)(329,347)
(330,348)(331,349)(332,350)(333,351)(334,352)(335,353)(336,354)(337,355)
(338,356)(339,357)(340,358)(341,359)(342,360)(361,379)(362,380)(363,381)
(364,382)(365,383)(366,384)(367,385)(368,386)(369,387)(370,388)(371,389)
(372,390)(373,391)(374,392)(375,393)(376,394)(377,395)(378,396)(397,415)
(398,416)(399,417)(400,418)(401,419)(402,420)(403,421)(404,422)(405,423)
(406,424)(407,425)(408,426)(409,427)(410,428)(411,429)(412,430)(413,431)
(414,432)(433,460)(434,461)(435,462)(436,463)(437,464)(438,465)(439,466)
(440,467)(441,468)(442,451)(443,452)(444,453)(445,454)(446,455)(447,456)
(448,457)(449,458)(450,459)(469,496)(470,497)(471,498)(472,499)(473,500)
(474,501)(475,502)(476,503)(477,504)(478,487)(479,488)(480,489)(481,490)
(482,491)(483,492)(484,493)(485,494)(486,495)(505,532)(506,533)(507,534)
(508,535)(509,536)(510,537)(511,538)(512,539)(513,540)(514,523)(515,524)
(516,525)(517,526)(518,527)(519,528)(520,529)(521,530)(522,531)(541,568)
(542,569)(543,570)(544,571)(545,572)(546,573)(547,574)(548,575)(549,576)
(550,559)(551,560)(552,561)(553,562)(554,563)(555,564)(556,565)(557,566)
(558,567);
s3 := Sym(576)!(  1,253)(  2,260)(  3,258)(  4,259)(  5,257)(  6,255)(  7,256)
(  8,254)(  9,261)( 10,262)( 11,269)( 12,267)( 13,268)( 14,266)( 15,264)
( 16,265)( 17,263)( 18,270)( 19,271)( 20,278)( 21,276)( 22,277)( 23,275)
( 24,273)( 25,274)( 26,272)( 27,279)( 28,280)( 29,287)( 30,285)( 31,286)
( 32,284)( 33,282)( 34,283)( 35,281)( 36,288)( 37,217)( 38,224)( 39,222)
( 40,223)( 41,221)( 42,219)( 43,220)( 44,218)( 45,225)( 46,226)( 47,233)
( 48,231)( 49,232)( 50,230)( 51,228)( 52,229)( 53,227)( 54,234)( 55,235)
( 56,242)( 57,240)( 58,241)( 59,239)( 60,237)( 61,238)( 62,236)( 63,243)
( 64,244)( 65,251)( 66,249)( 67,250)( 68,248)( 69,246)( 70,247)( 71,245)
( 72,252)( 73,181)( 74,188)( 75,186)( 76,187)( 77,185)( 78,183)( 79,184)
( 80,182)( 81,189)( 82,190)( 83,197)( 84,195)( 85,196)( 86,194)( 87,192)
( 88,193)( 89,191)( 90,198)( 91,199)( 92,206)( 93,204)( 94,205)( 95,203)
( 96,201)( 97,202)( 98,200)( 99,207)(100,208)(101,215)(102,213)(103,214)
(104,212)(105,210)(106,211)(107,209)(108,216)(109,145)(110,152)(111,150)
(112,151)(113,149)(114,147)(115,148)(116,146)(117,153)(118,154)(119,161)
(120,159)(121,160)(122,158)(123,156)(124,157)(125,155)(126,162)(127,163)
(128,170)(129,168)(130,169)(131,167)(132,165)(133,166)(134,164)(135,171)
(136,172)(137,179)(138,177)(139,178)(140,176)(141,174)(142,175)(143,173)
(144,180)(289,541)(290,548)(291,546)(292,547)(293,545)(294,543)(295,544)
(296,542)(297,549)(298,550)(299,557)(300,555)(301,556)(302,554)(303,552)
(304,553)(305,551)(306,558)(307,559)(308,566)(309,564)(310,565)(311,563)
(312,561)(313,562)(314,560)(315,567)(316,568)(317,575)(318,573)(319,574)
(320,572)(321,570)(322,571)(323,569)(324,576)(325,505)(326,512)(327,510)
(328,511)(329,509)(330,507)(331,508)(332,506)(333,513)(334,514)(335,521)
(336,519)(337,520)(338,518)(339,516)(340,517)(341,515)(342,522)(343,523)
(344,530)(345,528)(346,529)(347,527)(348,525)(349,526)(350,524)(351,531)
(352,532)(353,539)(354,537)(355,538)(356,536)(357,534)(358,535)(359,533)
(360,540)(361,469)(362,476)(363,474)(364,475)(365,473)(366,471)(367,472)
(368,470)(369,477)(370,478)(371,485)(372,483)(373,484)(374,482)(375,480)
(376,481)(377,479)(378,486)(379,487)(380,494)(381,492)(382,493)(383,491)
(384,489)(385,490)(386,488)(387,495)(388,496)(389,503)(390,501)(391,502)
(392,500)(393,498)(394,499)(395,497)(396,504)(397,433)(398,440)(399,438)
(400,439)(401,437)(402,435)(403,436)(404,434)(405,441)(406,442)(407,449)
(408,447)(409,448)(410,446)(411,444)(412,445)(413,443)(414,450)(415,451)
(416,458)(417,456)(418,457)(419,455)(420,453)(421,454)(422,452)(423,459)
(424,460)(425,467)(426,465)(427,466)(428,464)(429,462)(430,463)(431,461)
(432,468);
s4 := Sym(576)!(  1,112)(  2,110)(  3,117)(  4,109)(  5,116)(  6,114)(  7,115)
(  8,113)(  9,111)( 10,121)( 11,119)( 12,126)( 13,118)( 14,125)( 15,123)
( 16,124)( 17,122)( 18,120)( 19,130)( 20,128)( 21,135)( 22,127)( 23,134)
( 24,132)( 25,133)( 26,131)( 27,129)( 28,139)( 29,137)( 30,144)( 31,136)
( 32,143)( 33,141)( 34,142)( 35,140)( 36,138)( 37, 76)( 38, 74)( 39, 81)
( 40, 73)( 41, 80)( 42, 78)( 43, 79)( 44, 77)( 45, 75)( 46, 85)( 47, 83)
( 48, 90)( 49, 82)( 50, 89)( 51, 87)( 52, 88)( 53, 86)( 54, 84)( 55, 94)
( 56, 92)( 57, 99)( 58, 91)( 59, 98)( 60, 96)( 61, 97)( 62, 95)( 63, 93)
( 64,103)( 65,101)( 66,108)( 67,100)( 68,107)( 69,105)( 70,106)( 71,104)
( 72,102)(145,256)(146,254)(147,261)(148,253)(149,260)(150,258)(151,259)
(152,257)(153,255)(154,265)(155,263)(156,270)(157,262)(158,269)(159,267)
(160,268)(161,266)(162,264)(163,274)(164,272)(165,279)(166,271)(167,278)
(168,276)(169,277)(170,275)(171,273)(172,283)(173,281)(174,288)(175,280)
(176,287)(177,285)(178,286)(179,284)(180,282)(181,220)(182,218)(183,225)
(184,217)(185,224)(186,222)(187,223)(188,221)(189,219)(190,229)(191,227)
(192,234)(193,226)(194,233)(195,231)(196,232)(197,230)(198,228)(199,238)
(200,236)(201,243)(202,235)(203,242)(204,240)(205,241)(206,239)(207,237)
(208,247)(209,245)(210,252)(211,244)(212,251)(213,249)(214,250)(215,248)
(216,246)(289,400)(290,398)(291,405)(292,397)(293,404)(294,402)(295,403)
(296,401)(297,399)(298,409)(299,407)(300,414)(301,406)(302,413)(303,411)
(304,412)(305,410)(306,408)(307,418)(308,416)(309,423)(310,415)(311,422)
(312,420)(313,421)(314,419)(315,417)(316,427)(317,425)(318,432)(319,424)
(320,431)(321,429)(322,430)(323,428)(324,426)(325,364)(326,362)(327,369)
(328,361)(329,368)(330,366)(331,367)(332,365)(333,363)(334,373)(335,371)
(336,378)(337,370)(338,377)(339,375)(340,376)(341,374)(342,372)(343,382)
(344,380)(345,387)(346,379)(347,386)(348,384)(349,385)(350,383)(351,381)
(352,391)(353,389)(354,396)(355,388)(356,395)(357,393)(358,394)(359,392)
(360,390)(433,544)(434,542)(435,549)(436,541)(437,548)(438,546)(439,547)
(440,545)(441,543)(442,553)(443,551)(444,558)(445,550)(446,557)(447,555)
(448,556)(449,554)(450,552)(451,562)(452,560)(453,567)(454,559)(455,566)
(456,564)(457,565)(458,563)(459,561)(460,571)(461,569)(462,576)(463,568)
(464,575)(465,573)(466,574)(467,572)(468,570)(469,508)(470,506)(471,513)
(472,505)(473,512)(474,510)(475,511)(476,509)(477,507)(478,517)(479,515)
(480,522)(481,514)(482,521)(483,519)(484,520)(485,518)(486,516)(487,526)
(488,524)(489,531)(490,523)(491,530)(492,528)(493,529)(494,527)(495,525)
(496,535)(497,533)(498,540)(499,532)(500,539)(501,537)(502,538)(503,536)
(504,534);
poly := sub<Sym(576)|s0,s1,s2,s3,s4>;

Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3,s4> := Group< s0,s1,s2,s3,s4 | s0*s0, s1*s1, s2*s2,
s3*s3, s4*s4, s0*s2*s0*s2, s0*s3*s0*s3,
s1*s3*s1*s3, s0*s4*s0*s4, s1*s4*s1*s4,
s2*s4*s2*s4, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s2*s3*s4*s3*s2*s3*s4*s3,
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s3*s4*s3*s4*s3*s4*s3*s4*s3*s4*s3*s4 >;

References : None.
to this polytope