Polytope of Type {4,4,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,4,4}*1152a
Also Known As : 2T4(2,0)(6,0), {{4,4|2},{4,4|6}}. if this polytope has another name.
Group : SmallGroup(1152,43028)
Rank : 4
Schlafli Type : {4,4,4}
Number of vertices, edges, etc : 4, 72, 72, 36
Order of s0s1s2s3 : 12
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Locally Toroidal
Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,4,4}*576b, {2,4,4}*576
4-fold quotients : {2,4,4}*288
8-fold quotients : {2,4,4}*144
9-fold quotients : {4,4,4}*128
18-fold quotients : {2,4,4}*64, {4,4,2}*64, {4,2,4}*64
36-fold quotients : {2,2,4}*32, {2,4,2}*32, {4,2,2}*32
72-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1, 73)( 2, 74)( 3, 75)( 4, 76)( 5, 77)( 6, 78)( 7, 79)( 8, 80)
( 9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)( 16, 88)
( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)( 24, 96)
( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)( 32,104)
( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)( 40,112)
( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)( 48,120)
( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)( 56,128)
( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)( 64,136)
( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)( 72,144)
(145,217)(146,218)(147,219)(148,220)(149,221)(150,222)(151,223)(152,224)
(153,225)(154,226)(155,227)(156,228)(157,229)(158,230)(159,231)(160,232)
(161,233)(162,234)(163,235)(164,236)(165,237)(166,238)(167,239)(168,240)
(169,241)(170,242)(171,243)(172,244)(173,245)(174,246)(175,247)(176,248)
(177,249)(178,250)(179,251)(180,252)(181,253)(182,254)(183,255)(184,256)
(185,257)(186,258)(187,259)(188,260)(189,261)(190,262)(191,263)(192,264)
(193,265)(194,266)(195,267)(196,268)(197,269)(198,270)(199,271)(200,272)
(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)(208,280)
(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)(216,288)
(289,361)(290,362)(291,363)(292,364)(293,365)(294,366)(295,367)(296,368)
(297,369)(298,370)(299,371)(300,372)(301,373)(302,374)(303,375)(304,376)
(305,377)(306,378)(307,379)(308,380)(309,381)(310,382)(311,383)(312,384)
(313,385)(314,386)(315,387)(316,388)(317,389)(318,390)(319,391)(320,392)
(321,393)(322,394)(323,395)(324,396)(325,397)(326,398)(327,399)(328,400)
(329,401)(330,402)(331,403)(332,404)(333,405)(334,406)(335,407)(336,408)
(337,409)(338,410)(339,411)(340,412)(341,413)(342,414)(343,415)(344,416)
(345,417)(346,418)(347,419)(348,420)(349,421)(350,422)(351,423)(352,424)
(353,425)(354,426)(355,427)(356,428)(357,429)(358,430)(359,431)(360,432)
(433,505)(434,506)(435,507)(436,508)(437,509)(438,510)(439,511)(440,512)
(441,513)(442,514)(443,515)(444,516)(445,517)(446,518)(447,519)(448,520)
(449,521)(450,522)(451,523)(452,524)(453,525)(454,526)(455,527)(456,528)
(457,529)(458,530)(459,531)(460,532)(461,533)(462,534)(463,535)(464,536)
(465,537)(466,538)(467,539)(468,540)(469,541)(470,542)(471,543)(472,544)
(473,545)(474,546)(475,547)(476,548)(477,549)(478,550)(479,551)(480,552)
(481,553)(482,554)(483,555)(484,556)(485,557)(486,558)(487,559)(488,560)
(489,561)(490,562)(491,563)(492,564)(493,565)(494,566)(495,567)(496,568)
(497,569)(498,570)(499,571)(500,572)(501,573)(502,574)(503,575)(504,576);;
s1 := ( 1,361)( 2,366)( 3,368)( 4,364)( 5,369)( 6,362)( 7,367)( 8,363)
( 9,365)( 10,370)( 11,375)( 12,377)( 13,373)( 14,378)( 15,371)( 16,376)
( 17,372)( 18,374)( 19,379)( 20,384)( 21,386)( 22,382)( 23,387)( 24,380)
( 25,385)( 26,381)( 27,383)( 28,388)( 29,393)( 30,395)( 31,391)( 32,396)
( 33,389)( 34,394)( 35,390)( 36,392)( 37,397)( 38,402)( 39,404)( 40,400)
( 41,405)( 42,398)( 43,403)( 44,399)( 45,401)( 46,406)( 47,411)( 48,413)
( 49,409)( 50,414)( 51,407)( 52,412)( 53,408)( 54,410)( 55,415)( 56,420)
( 57,422)( 58,418)( 59,423)( 60,416)( 61,421)( 62,417)( 63,419)( 64,424)
( 65,429)( 66,431)( 67,427)( 68,432)( 69,425)( 70,430)( 71,426)( 72,428)
( 73,325)( 74,330)( 75,332)( 76,328)( 77,333)( 78,326)( 79,331)( 80,327)
( 81,329)( 82,334)( 83,339)( 84,341)( 85,337)( 86,342)( 87,335)( 88,340)
( 89,336)( 90,338)( 91,343)( 92,348)( 93,350)( 94,346)( 95,351)( 96,344)
( 97,349)( 98,345)( 99,347)(100,352)(101,357)(102,359)(103,355)(104,360)
(105,353)(106,358)(107,354)(108,356)(109,289)(110,294)(111,296)(112,292)
(113,297)(114,290)(115,295)(116,291)(117,293)(118,298)(119,303)(120,305)
(121,301)(122,306)(123,299)(124,304)(125,300)(126,302)(127,307)(128,312)
(129,314)(130,310)(131,315)(132,308)(133,313)(134,309)(135,311)(136,316)
(137,321)(138,323)(139,319)(140,324)(141,317)(142,322)(143,318)(144,320)
(145,532)(146,537)(147,539)(148,535)(149,540)(150,533)(151,538)(152,534)
(153,536)(154,523)(155,528)(156,530)(157,526)(158,531)(159,524)(160,529)
(161,525)(162,527)(163,514)(164,519)(165,521)(166,517)(167,522)(168,515)
(169,520)(170,516)(171,518)(172,505)(173,510)(174,512)(175,508)(176,513)
(177,506)(178,511)(179,507)(180,509)(181,568)(182,573)(183,575)(184,571)
(185,576)(186,569)(187,574)(188,570)(189,572)(190,559)(191,564)(192,566)
(193,562)(194,567)(195,560)(196,565)(197,561)(198,563)(199,550)(200,555)
(201,557)(202,553)(203,558)(204,551)(205,556)(206,552)(207,554)(208,541)
(209,546)(210,548)(211,544)(212,549)(213,542)(214,547)(215,543)(216,545)
(217,496)(218,501)(219,503)(220,499)(221,504)(222,497)(223,502)(224,498)
(225,500)(226,487)(227,492)(228,494)(229,490)(230,495)(231,488)(232,493)
(233,489)(234,491)(235,478)(236,483)(237,485)(238,481)(239,486)(240,479)
(241,484)(242,480)(243,482)(244,469)(245,474)(246,476)(247,472)(248,477)
(249,470)(250,475)(251,471)(252,473)(253,460)(254,465)(255,467)(256,463)
(257,468)(258,461)(259,466)(260,462)(261,464)(262,451)(263,456)(264,458)
(265,454)(266,459)(267,452)(268,457)(269,453)(270,455)(271,442)(272,447)
(273,449)(274,445)(275,450)(276,443)(277,448)(278,444)(279,446)(280,433)
(281,438)(282,440)(283,436)(284,441)(285,434)(286,439)(287,435)(288,437);;
s2 := ( 1,145)( 2,147)( 3,146)( 4,149)( 5,148)( 6,150)( 7,153)( 8,152)
( 9,151)( 10,154)( 11,156)( 12,155)( 13,158)( 14,157)( 15,159)( 16,162)
( 17,161)( 18,160)( 19,163)( 20,165)( 21,164)( 22,167)( 23,166)( 24,168)
( 25,171)( 26,170)( 27,169)( 28,172)( 29,174)( 30,173)( 31,176)( 32,175)
( 33,177)( 34,180)( 35,179)( 36,178)( 37,181)( 38,183)( 39,182)( 40,185)
( 41,184)( 42,186)( 43,189)( 44,188)( 45,187)( 46,190)( 47,192)( 48,191)
( 49,194)( 50,193)( 51,195)( 52,198)( 53,197)( 54,196)( 55,199)( 56,201)
( 57,200)( 58,203)( 59,202)( 60,204)( 61,207)( 62,206)( 63,205)( 64,208)
( 65,210)( 66,209)( 67,212)( 68,211)( 69,213)( 70,216)( 71,215)( 72,214)
( 73,217)( 74,219)( 75,218)( 76,221)( 77,220)( 78,222)( 79,225)( 80,224)
( 81,223)( 82,226)( 83,228)( 84,227)( 85,230)( 86,229)( 87,231)( 88,234)
( 89,233)( 90,232)( 91,235)( 92,237)( 93,236)( 94,239)( 95,238)( 96,240)
( 97,243)( 98,242)( 99,241)(100,244)(101,246)(102,245)(103,248)(104,247)
(105,249)(106,252)(107,251)(108,250)(109,253)(110,255)(111,254)(112,257)
(113,256)(114,258)(115,261)(116,260)(117,259)(118,262)(119,264)(120,263)
(121,266)(122,265)(123,267)(124,270)(125,269)(126,268)(127,271)(128,273)
(129,272)(130,275)(131,274)(132,276)(133,279)(134,278)(135,277)(136,280)
(137,282)(138,281)(139,284)(140,283)(141,285)(142,288)(143,287)(144,286)
(289,433)(290,435)(291,434)(292,437)(293,436)(294,438)(295,441)(296,440)
(297,439)(298,442)(299,444)(300,443)(301,446)(302,445)(303,447)(304,450)
(305,449)(306,448)(307,451)(308,453)(309,452)(310,455)(311,454)(312,456)
(313,459)(314,458)(315,457)(316,460)(317,462)(318,461)(319,464)(320,463)
(321,465)(322,468)(323,467)(324,466)(325,469)(326,471)(327,470)(328,473)
(329,472)(330,474)(331,477)(332,476)(333,475)(334,478)(335,480)(336,479)
(337,482)(338,481)(339,483)(340,486)(341,485)(342,484)(343,487)(344,489)
(345,488)(346,491)(347,490)(348,492)(349,495)(350,494)(351,493)(352,496)
(353,498)(354,497)(355,500)(356,499)(357,501)(358,504)(359,503)(360,502)
(361,505)(362,507)(363,506)(364,509)(365,508)(366,510)(367,513)(368,512)
(369,511)(370,514)(371,516)(372,515)(373,518)(374,517)(375,519)(376,522)
(377,521)(378,520)(379,523)(380,525)(381,524)(382,527)(383,526)(384,528)
(385,531)(386,530)(387,529)(388,532)(389,534)(390,533)(391,536)(392,535)
(393,537)(394,540)(395,539)(396,538)(397,541)(398,543)(399,542)(400,545)
(401,544)(402,546)(403,549)(404,548)(405,547)(406,550)(407,552)(408,551)
(409,554)(410,553)(411,555)(412,558)(413,557)(414,556)(415,559)(416,561)
(417,560)(418,563)(419,562)(420,564)(421,567)(422,566)(423,565)(424,568)
(425,570)(426,569)(427,572)(428,571)(429,573)(430,576)(431,575)(432,574);;
s3 := ( 1,373)( 2,371)( 3,378)( 4,370)( 5,377)( 6,375)( 7,376)( 8,374)
( 9,372)( 10,364)( 11,362)( 12,369)( 13,361)( 14,368)( 15,366)( 16,367)
( 17,365)( 18,363)( 19,391)( 20,389)( 21,396)( 22,388)( 23,395)( 24,393)
( 25,394)( 26,392)( 27,390)( 28,382)( 29,380)( 30,387)( 31,379)( 32,386)
( 33,384)( 34,385)( 35,383)( 36,381)( 37,409)( 38,407)( 39,414)( 40,406)
( 41,413)( 42,411)( 43,412)( 44,410)( 45,408)( 46,400)( 47,398)( 48,405)
( 49,397)( 50,404)( 51,402)( 52,403)( 53,401)( 54,399)( 55,427)( 56,425)
( 57,432)( 58,424)( 59,431)( 60,429)( 61,430)( 62,428)( 63,426)( 64,418)
( 65,416)( 66,423)( 67,415)( 68,422)( 69,420)( 70,421)( 71,419)( 72,417)
( 73,301)( 74,299)( 75,306)( 76,298)( 77,305)( 78,303)( 79,304)( 80,302)
( 81,300)( 82,292)( 83,290)( 84,297)( 85,289)( 86,296)( 87,294)( 88,295)
( 89,293)( 90,291)( 91,319)( 92,317)( 93,324)( 94,316)( 95,323)( 96,321)
( 97,322)( 98,320)( 99,318)(100,310)(101,308)(102,315)(103,307)(104,314)
(105,312)(106,313)(107,311)(108,309)(109,337)(110,335)(111,342)(112,334)
(113,341)(114,339)(115,340)(116,338)(117,336)(118,328)(119,326)(120,333)
(121,325)(122,332)(123,330)(124,331)(125,329)(126,327)(127,355)(128,353)
(129,360)(130,352)(131,359)(132,357)(133,358)(134,356)(135,354)(136,346)
(137,344)(138,351)(139,343)(140,350)(141,348)(142,349)(143,347)(144,345)
(145,508)(146,506)(147,513)(148,505)(149,512)(150,510)(151,511)(152,509)
(153,507)(154,517)(155,515)(156,522)(157,514)(158,521)(159,519)(160,520)
(161,518)(162,516)(163,526)(164,524)(165,531)(166,523)(167,530)(168,528)
(169,529)(170,527)(171,525)(172,535)(173,533)(174,540)(175,532)(176,539)
(177,537)(178,538)(179,536)(180,534)(181,544)(182,542)(183,549)(184,541)
(185,548)(186,546)(187,547)(188,545)(189,543)(190,553)(191,551)(192,558)
(193,550)(194,557)(195,555)(196,556)(197,554)(198,552)(199,562)(200,560)
(201,567)(202,559)(203,566)(204,564)(205,565)(206,563)(207,561)(208,571)
(209,569)(210,576)(211,568)(212,575)(213,573)(214,574)(215,572)(216,570)
(217,436)(218,434)(219,441)(220,433)(221,440)(222,438)(223,439)(224,437)
(225,435)(226,445)(227,443)(228,450)(229,442)(230,449)(231,447)(232,448)
(233,446)(234,444)(235,454)(236,452)(237,459)(238,451)(239,458)(240,456)
(241,457)(242,455)(243,453)(244,463)(245,461)(246,468)(247,460)(248,467)
(249,465)(250,466)(251,464)(252,462)(253,472)(254,470)(255,477)(256,469)
(257,476)(258,474)(259,475)(260,473)(261,471)(262,481)(263,479)(264,486)
(265,478)(266,485)(267,483)(268,484)(269,482)(270,480)(271,490)(272,488)
(273,495)(274,487)(275,494)(276,492)(277,493)(278,491)(279,489)(280,499)
(281,497)(282,504)(283,496)(284,503)(285,501)(286,502)(287,500)(288,498);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3, s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(576)!( 1, 73)( 2, 74)( 3, 75)( 4, 76)( 5, 77)( 6, 78)( 7, 79)
( 8, 80)( 9, 81)( 10, 82)( 11, 83)( 12, 84)( 13, 85)( 14, 86)( 15, 87)
( 16, 88)( 17, 89)( 18, 90)( 19, 91)( 20, 92)( 21, 93)( 22, 94)( 23, 95)
( 24, 96)( 25, 97)( 26, 98)( 27, 99)( 28,100)( 29,101)( 30,102)( 31,103)
( 32,104)( 33,105)( 34,106)( 35,107)( 36,108)( 37,109)( 38,110)( 39,111)
( 40,112)( 41,113)( 42,114)( 43,115)( 44,116)( 45,117)( 46,118)( 47,119)
( 48,120)( 49,121)( 50,122)( 51,123)( 52,124)( 53,125)( 54,126)( 55,127)
( 56,128)( 57,129)( 58,130)( 59,131)( 60,132)( 61,133)( 62,134)( 63,135)
( 64,136)( 65,137)( 66,138)( 67,139)( 68,140)( 69,141)( 70,142)( 71,143)
( 72,144)(145,217)(146,218)(147,219)(148,220)(149,221)(150,222)(151,223)
(152,224)(153,225)(154,226)(155,227)(156,228)(157,229)(158,230)(159,231)
(160,232)(161,233)(162,234)(163,235)(164,236)(165,237)(166,238)(167,239)
(168,240)(169,241)(170,242)(171,243)(172,244)(173,245)(174,246)(175,247)
(176,248)(177,249)(178,250)(179,251)(180,252)(181,253)(182,254)(183,255)
(184,256)(185,257)(186,258)(187,259)(188,260)(189,261)(190,262)(191,263)
(192,264)(193,265)(194,266)(195,267)(196,268)(197,269)(198,270)(199,271)
(200,272)(201,273)(202,274)(203,275)(204,276)(205,277)(206,278)(207,279)
(208,280)(209,281)(210,282)(211,283)(212,284)(213,285)(214,286)(215,287)
(216,288)(289,361)(290,362)(291,363)(292,364)(293,365)(294,366)(295,367)
(296,368)(297,369)(298,370)(299,371)(300,372)(301,373)(302,374)(303,375)
(304,376)(305,377)(306,378)(307,379)(308,380)(309,381)(310,382)(311,383)
(312,384)(313,385)(314,386)(315,387)(316,388)(317,389)(318,390)(319,391)
(320,392)(321,393)(322,394)(323,395)(324,396)(325,397)(326,398)(327,399)
(328,400)(329,401)(330,402)(331,403)(332,404)(333,405)(334,406)(335,407)
(336,408)(337,409)(338,410)(339,411)(340,412)(341,413)(342,414)(343,415)
(344,416)(345,417)(346,418)(347,419)(348,420)(349,421)(350,422)(351,423)
(352,424)(353,425)(354,426)(355,427)(356,428)(357,429)(358,430)(359,431)
(360,432)(433,505)(434,506)(435,507)(436,508)(437,509)(438,510)(439,511)
(440,512)(441,513)(442,514)(443,515)(444,516)(445,517)(446,518)(447,519)
(448,520)(449,521)(450,522)(451,523)(452,524)(453,525)(454,526)(455,527)
(456,528)(457,529)(458,530)(459,531)(460,532)(461,533)(462,534)(463,535)
(464,536)(465,537)(466,538)(467,539)(468,540)(469,541)(470,542)(471,543)
(472,544)(473,545)(474,546)(475,547)(476,548)(477,549)(478,550)(479,551)
(480,552)(481,553)(482,554)(483,555)(484,556)(485,557)(486,558)(487,559)
(488,560)(489,561)(490,562)(491,563)(492,564)(493,565)(494,566)(495,567)
(496,568)(497,569)(498,570)(499,571)(500,572)(501,573)(502,574)(503,575)
(504,576);
s1 := Sym(576)!( 1,361)( 2,366)( 3,368)( 4,364)( 5,369)( 6,362)( 7,367)
( 8,363)( 9,365)( 10,370)( 11,375)( 12,377)( 13,373)( 14,378)( 15,371)
( 16,376)( 17,372)( 18,374)( 19,379)( 20,384)( 21,386)( 22,382)( 23,387)
( 24,380)( 25,385)( 26,381)( 27,383)( 28,388)( 29,393)( 30,395)( 31,391)
( 32,396)( 33,389)( 34,394)( 35,390)( 36,392)( 37,397)( 38,402)( 39,404)
( 40,400)( 41,405)( 42,398)( 43,403)( 44,399)( 45,401)( 46,406)( 47,411)
( 48,413)( 49,409)( 50,414)( 51,407)( 52,412)( 53,408)( 54,410)( 55,415)
( 56,420)( 57,422)( 58,418)( 59,423)( 60,416)( 61,421)( 62,417)( 63,419)
( 64,424)( 65,429)( 66,431)( 67,427)( 68,432)( 69,425)( 70,430)( 71,426)
( 72,428)( 73,325)( 74,330)( 75,332)( 76,328)( 77,333)( 78,326)( 79,331)
( 80,327)( 81,329)( 82,334)( 83,339)( 84,341)( 85,337)( 86,342)( 87,335)
( 88,340)( 89,336)( 90,338)( 91,343)( 92,348)( 93,350)( 94,346)( 95,351)
( 96,344)( 97,349)( 98,345)( 99,347)(100,352)(101,357)(102,359)(103,355)
(104,360)(105,353)(106,358)(107,354)(108,356)(109,289)(110,294)(111,296)
(112,292)(113,297)(114,290)(115,295)(116,291)(117,293)(118,298)(119,303)
(120,305)(121,301)(122,306)(123,299)(124,304)(125,300)(126,302)(127,307)
(128,312)(129,314)(130,310)(131,315)(132,308)(133,313)(134,309)(135,311)
(136,316)(137,321)(138,323)(139,319)(140,324)(141,317)(142,322)(143,318)
(144,320)(145,532)(146,537)(147,539)(148,535)(149,540)(150,533)(151,538)
(152,534)(153,536)(154,523)(155,528)(156,530)(157,526)(158,531)(159,524)
(160,529)(161,525)(162,527)(163,514)(164,519)(165,521)(166,517)(167,522)
(168,515)(169,520)(170,516)(171,518)(172,505)(173,510)(174,512)(175,508)
(176,513)(177,506)(178,511)(179,507)(180,509)(181,568)(182,573)(183,575)
(184,571)(185,576)(186,569)(187,574)(188,570)(189,572)(190,559)(191,564)
(192,566)(193,562)(194,567)(195,560)(196,565)(197,561)(198,563)(199,550)
(200,555)(201,557)(202,553)(203,558)(204,551)(205,556)(206,552)(207,554)
(208,541)(209,546)(210,548)(211,544)(212,549)(213,542)(214,547)(215,543)
(216,545)(217,496)(218,501)(219,503)(220,499)(221,504)(222,497)(223,502)
(224,498)(225,500)(226,487)(227,492)(228,494)(229,490)(230,495)(231,488)
(232,493)(233,489)(234,491)(235,478)(236,483)(237,485)(238,481)(239,486)
(240,479)(241,484)(242,480)(243,482)(244,469)(245,474)(246,476)(247,472)
(248,477)(249,470)(250,475)(251,471)(252,473)(253,460)(254,465)(255,467)
(256,463)(257,468)(258,461)(259,466)(260,462)(261,464)(262,451)(263,456)
(264,458)(265,454)(266,459)(267,452)(268,457)(269,453)(270,455)(271,442)
(272,447)(273,449)(274,445)(275,450)(276,443)(277,448)(278,444)(279,446)
(280,433)(281,438)(282,440)(283,436)(284,441)(285,434)(286,439)(287,435)
(288,437);
s2 := Sym(576)!( 1,145)( 2,147)( 3,146)( 4,149)( 5,148)( 6,150)( 7,153)
( 8,152)( 9,151)( 10,154)( 11,156)( 12,155)( 13,158)( 14,157)( 15,159)
( 16,162)( 17,161)( 18,160)( 19,163)( 20,165)( 21,164)( 22,167)( 23,166)
( 24,168)( 25,171)( 26,170)( 27,169)( 28,172)( 29,174)( 30,173)( 31,176)
( 32,175)( 33,177)( 34,180)( 35,179)( 36,178)( 37,181)( 38,183)( 39,182)
( 40,185)( 41,184)( 42,186)( 43,189)( 44,188)( 45,187)( 46,190)( 47,192)
( 48,191)( 49,194)( 50,193)( 51,195)( 52,198)( 53,197)( 54,196)( 55,199)
( 56,201)( 57,200)( 58,203)( 59,202)( 60,204)( 61,207)( 62,206)( 63,205)
( 64,208)( 65,210)( 66,209)( 67,212)( 68,211)( 69,213)( 70,216)( 71,215)
( 72,214)( 73,217)( 74,219)( 75,218)( 76,221)( 77,220)( 78,222)( 79,225)
( 80,224)( 81,223)( 82,226)( 83,228)( 84,227)( 85,230)( 86,229)( 87,231)
( 88,234)( 89,233)( 90,232)( 91,235)( 92,237)( 93,236)( 94,239)( 95,238)
( 96,240)( 97,243)( 98,242)( 99,241)(100,244)(101,246)(102,245)(103,248)
(104,247)(105,249)(106,252)(107,251)(108,250)(109,253)(110,255)(111,254)
(112,257)(113,256)(114,258)(115,261)(116,260)(117,259)(118,262)(119,264)
(120,263)(121,266)(122,265)(123,267)(124,270)(125,269)(126,268)(127,271)
(128,273)(129,272)(130,275)(131,274)(132,276)(133,279)(134,278)(135,277)
(136,280)(137,282)(138,281)(139,284)(140,283)(141,285)(142,288)(143,287)
(144,286)(289,433)(290,435)(291,434)(292,437)(293,436)(294,438)(295,441)
(296,440)(297,439)(298,442)(299,444)(300,443)(301,446)(302,445)(303,447)
(304,450)(305,449)(306,448)(307,451)(308,453)(309,452)(310,455)(311,454)
(312,456)(313,459)(314,458)(315,457)(316,460)(317,462)(318,461)(319,464)
(320,463)(321,465)(322,468)(323,467)(324,466)(325,469)(326,471)(327,470)
(328,473)(329,472)(330,474)(331,477)(332,476)(333,475)(334,478)(335,480)
(336,479)(337,482)(338,481)(339,483)(340,486)(341,485)(342,484)(343,487)
(344,489)(345,488)(346,491)(347,490)(348,492)(349,495)(350,494)(351,493)
(352,496)(353,498)(354,497)(355,500)(356,499)(357,501)(358,504)(359,503)
(360,502)(361,505)(362,507)(363,506)(364,509)(365,508)(366,510)(367,513)
(368,512)(369,511)(370,514)(371,516)(372,515)(373,518)(374,517)(375,519)
(376,522)(377,521)(378,520)(379,523)(380,525)(381,524)(382,527)(383,526)
(384,528)(385,531)(386,530)(387,529)(388,532)(389,534)(390,533)(391,536)
(392,535)(393,537)(394,540)(395,539)(396,538)(397,541)(398,543)(399,542)
(400,545)(401,544)(402,546)(403,549)(404,548)(405,547)(406,550)(407,552)
(408,551)(409,554)(410,553)(411,555)(412,558)(413,557)(414,556)(415,559)
(416,561)(417,560)(418,563)(419,562)(420,564)(421,567)(422,566)(423,565)
(424,568)(425,570)(426,569)(427,572)(428,571)(429,573)(430,576)(431,575)
(432,574);
s3 := Sym(576)!( 1,373)( 2,371)( 3,378)( 4,370)( 5,377)( 6,375)( 7,376)
( 8,374)( 9,372)( 10,364)( 11,362)( 12,369)( 13,361)( 14,368)( 15,366)
( 16,367)( 17,365)( 18,363)( 19,391)( 20,389)( 21,396)( 22,388)( 23,395)
( 24,393)( 25,394)( 26,392)( 27,390)( 28,382)( 29,380)( 30,387)( 31,379)
( 32,386)( 33,384)( 34,385)( 35,383)( 36,381)( 37,409)( 38,407)( 39,414)
( 40,406)( 41,413)( 42,411)( 43,412)( 44,410)( 45,408)( 46,400)( 47,398)
( 48,405)( 49,397)( 50,404)( 51,402)( 52,403)( 53,401)( 54,399)( 55,427)
( 56,425)( 57,432)( 58,424)( 59,431)( 60,429)( 61,430)( 62,428)( 63,426)
( 64,418)( 65,416)( 66,423)( 67,415)( 68,422)( 69,420)( 70,421)( 71,419)
( 72,417)( 73,301)( 74,299)( 75,306)( 76,298)( 77,305)( 78,303)( 79,304)
( 80,302)( 81,300)( 82,292)( 83,290)( 84,297)( 85,289)( 86,296)( 87,294)
( 88,295)( 89,293)( 90,291)( 91,319)( 92,317)( 93,324)( 94,316)( 95,323)
( 96,321)( 97,322)( 98,320)( 99,318)(100,310)(101,308)(102,315)(103,307)
(104,314)(105,312)(106,313)(107,311)(108,309)(109,337)(110,335)(111,342)
(112,334)(113,341)(114,339)(115,340)(116,338)(117,336)(118,328)(119,326)
(120,333)(121,325)(122,332)(123,330)(124,331)(125,329)(126,327)(127,355)
(128,353)(129,360)(130,352)(131,359)(132,357)(133,358)(134,356)(135,354)
(136,346)(137,344)(138,351)(139,343)(140,350)(141,348)(142,349)(143,347)
(144,345)(145,508)(146,506)(147,513)(148,505)(149,512)(150,510)(151,511)
(152,509)(153,507)(154,517)(155,515)(156,522)(157,514)(158,521)(159,519)
(160,520)(161,518)(162,516)(163,526)(164,524)(165,531)(166,523)(167,530)
(168,528)(169,529)(170,527)(171,525)(172,535)(173,533)(174,540)(175,532)
(176,539)(177,537)(178,538)(179,536)(180,534)(181,544)(182,542)(183,549)
(184,541)(185,548)(186,546)(187,547)(188,545)(189,543)(190,553)(191,551)
(192,558)(193,550)(194,557)(195,555)(196,556)(197,554)(198,552)(199,562)
(200,560)(201,567)(202,559)(203,566)(204,564)(205,565)(206,563)(207,561)
(208,571)(209,569)(210,576)(211,568)(212,575)(213,573)(214,574)(215,572)
(216,570)(217,436)(218,434)(219,441)(220,433)(221,440)(222,438)(223,439)
(224,437)(225,435)(226,445)(227,443)(228,450)(229,442)(230,449)(231,447)
(232,448)(233,446)(234,444)(235,454)(236,452)(237,459)(238,451)(239,458)
(240,456)(241,457)(242,455)(243,453)(244,463)(245,461)(246,468)(247,460)
(248,467)(249,465)(250,466)(251,464)(252,462)(253,472)(254,470)(255,477)
(256,469)(257,476)(258,474)(259,475)(260,473)(261,471)(262,481)(263,479)
(264,486)(265,478)(266,485)(267,483)(268,484)(269,482)(270,480)(271,490)
(272,488)(273,495)(274,487)(275,494)(276,492)(277,493)(278,491)(279,489)
(280,499)(281,497)(282,504)(283,496)(284,503)(285,501)(286,502)(287,500)
(288,498);
poly := sub<Sym(576)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2, s2*s3*s2*s3*s2*s3*s2*s3,
s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2*s1*s2*s3*s2 >;
References : - Theorem 10C12, McMullen P., Schulte, E.; Abstract Regular Polytopes (Camb\
ridge University Press, 2002)
to this polytope