Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,4,4}

Atlas Canonical Name {4,4,4}*1152b

Overview

Group
SmallGroup(1152,43028)
Rank
4
Schläfli Type
{4,4,4}
Vertices, edges, …
36, 72, 72, 4
Order of s0s1s2s3
12
Order of s0s1s2s3s2s1
2
Also known as
2T4(6,0)(2,0), {{4,4|6},{4,4|2}}. if this polytope has another name.

Special Properties

  • Universal
  • Locally Toroidal
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

4-fold

8-fold

9-fold

18-fold

36-fold

72-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s1*(s2*s1*s0)^4*s2*s1*s2> of order 2

4 facets

  • 4 of 2-fold non-regular quotient of {4,4}*288

20 vertex figures

P/N, where N=<(s1*s0*s2)^5*s1*s2> of order 2

4 facets

  • 4 of 2-fold non-regular quotient of {4,4}*288

18 vertex figures

P/N, where N=<(s0*s1)^2> of order 2

4 facets

  • 4 of 2-fold non-regular quotient of {4,4}*288

18 vertex figures

P/N, where N=<(s0*s2*s1)^4> of order 3

4 facets

  • 4 of 3-fold non-regular quotient of {4,4}*288

12 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2> of order 3

4 facets

  • 4 of 3-fold non-regular quotient of {4,4}*288

12 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^4*s2*s1*s2> of order 4

4 facets

  • 4 of 4-fold non-regular quotient of {4,4}*288

9 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s0*s1*s2*s1)^2*s0*(s1*s2)^2> of order 6

4 facets

  • 4 of 6-fold non-regular quotient of {4,4}*288

6 vertex figures

P/N, where N=<s0*s1*s2*s1*s0*s2, s1*s0*s1*s2*s1*s0*s2*s1> of order 6

4 facets

  • 4 of 6-fold non-regular quotient of {4,4}*288

8 vertex figures

P/N, where N=<(s0*s1)^2, s0*s1*(s2*s1*s0)^3*s2> of order 6

4 facets

  • 4 of 6-fold non-regular quotient of {4,4}*288

6 vertex figures

P/N, where N=<(s0*s1)^2, s0*s2*s1*s0*s1*s2> of order 6

4 facets

  • 4 of 6-fold non-regular quotient of {4,4}*288

6 vertex figures

P/N, where N=<(s1*s0*s1*s2)^2, (s1*s2*s1*s0)^2*(s1*s2)^2> of order 6

4 facets

  • 4 of 6-fold non-regular quotient of {4,4}*288

8 vertex figures

Representations

Permutation Representation (GAP) 
s0 := (  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)( 73,289)( 74,294)( 75,296)( 76,292)( 77,297)( 78,290)( 79,295)( 80,291)( 81,293)( 82,298)( 83,303)( 84,305)( 85,301)( 86,306)( 87,299)( 88,304)( 89,300)( 90,302)( 91,307)( 92,312)( 93,314)( 94,310)( 95,315)( 96,308)( 97,313)( 98,309)( 99,311)(100,316)(101,321)(102,323)(103,319)(104,324)(105,317)(106,322)(107,318)(108,320)(109,325)(110,330)(111,332)(112,328)(113,333)(114,326)(115,331)(116,327)(117,329)(118,334)(119,339)(120,341)(121,337)(122,342)(123,335)(124,340)(125,336)(126,338)(127,343)(128,348)(129,350)(130,346)(131,351)(132,344)(133,349)(134,345)(135,347)(136,352)(137,357)(138,359)(139,355)(140,360)(141,353)(142,358)(143,354)(144,356)(145,514)(146,519)(147,521)(148,517)(149,522)(150,515)(151,520)(152,516)(153,518)(154,505)(155,510)(156,512)(157,508)(158,513)(159,506)(160,511)(161,507)(162,509)(163,532)(164,537)(165,539)(166,535)(167,540)(168,533)(169,538)(170,534)(171,536)(172,523)(173,528)(174,530)(175,526)(176,531)(177,524)(178,529)(179,525)(180,527)(181,550)(182,555)(183,557)(184,553)(185,558)(186,551)(187,556)(188,552)(189,554)(190,541)(191,546)(192,548)(193,544)(194,549)(195,542)(196,547)(197,543)(198,545)(199,568)(200,573)(201,575)(202,571)(203,576)(204,569)(205,574)(206,570)(207,572)(208,559)(209,564)(210,566)(211,562)(212,567)(213,560)(214,565)(215,561)(216,563)(217,442)(218,447)(219,449)(220,445)(221,450)(222,443)(223,448)(224,444)(225,446)(226,433)(227,438)(228,440)(229,436)(230,441)(231,434)(232,439)(233,435)(234,437)(235,460)(236,465)(237,467)(238,463)(239,468)(240,461)(241,466)(242,462)(243,464)(244,451)(245,456)(246,458)(247,454)(248,459)(249,452)(250,457)(251,453)(252,455)(253,478)(254,483)(255,485)(256,481)(257,486)(258,479)(259,484)(260,480)(261,482)(262,469)(263,474)(264,476)(265,472)(266,477)(267,470)(268,475)(269,471)(270,473)(271,496)(272,501)(273,503)(274,499)(275,504)(276,497)(277,502)(278,498)(279,500)(280,487)(281,492)(282,494)(283,490)(284,495)(285,488)(286,493)(287,489)(288,491);;
s1 := (  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);;
s2 := (  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)( 73,337)( 74,335)( 75,342)( 76,334)( 77,341)( 78,339)( 79,340)( 80,338)( 81,336)( 82,328)( 83,326)( 84,333)( 85,325)( 86,332)( 87,330)( 88,331)( 89,329)( 90,327)( 91,355)( 92,353)( 93,360)( 94,352)( 95,359)( 96,357)( 97,358)( 98,356)( 99,354)(100,346)(101,344)(102,351)(103,343)(104,350)(105,348)(106,349)(107,347)(108,345)(109,301)(110,299)(111,306)(112,298)(113,305)(114,303)(115,304)(116,302)(117,300)(118,292)(119,290)(120,297)(121,289)(122,296)(123,294)(124,295)(125,293)(126,291)(127,319)(128,317)(129,324)(130,316)(131,323)(132,321)(133,322)(134,320)(135,318)(136,310)(137,308)(138,315)(139,307)(140,314)(141,312)(142,313)(143,311)(144,309)(145,526)(146,524)(147,531)(148,523)(149,530)(150,528)(151,529)(152,527)(153,525)(154,535)(155,533)(156,540)(157,532)(158,539)(159,537)(160,538)(161,536)(162,534)(163,508)(164,506)(165,513)(166,505)(167,512)(168,510)(169,511)(170,509)(171,507)(172,517)(173,515)(174,522)(175,514)(176,521)(177,519)(178,520)(179,518)(180,516)(181,562)(182,560)(183,567)(184,559)(185,566)(186,564)(187,565)(188,563)(189,561)(190,571)(191,569)(192,576)(193,568)(194,575)(195,573)(196,574)(197,572)(198,570)(199,544)(200,542)(201,549)(202,541)(203,548)(204,546)(205,547)(206,545)(207,543)(208,553)(209,551)(210,558)(211,550)(212,557)(213,555)(214,556)(215,554)(216,552)(217,490)(218,488)(219,495)(220,487)(221,494)(222,492)(223,493)(224,491)(225,489)(226,499)(227,497)(228,504)(229,496)(230,503)(231,501)(232,502)(233,500)(234,498)(235,472)(236,470)(237,477)(238,469)(239,476)(240,474)(241,475)(242,473)(243,471)(244,481)(245,479)(246,486)(247,478)(248,485)(249,483)(250,484)(251,482)(252,480)(253,454)(254,452)(255,459)(256,451)(257,458)(258,456)(259,457)(260,455)(261,453)(262,463)(263,461)(264,468)(265,460)(266,467)(267,465)(268,466)(269,464)(270,462)(271,436)(272,434)(273,441)(274,433)(275,440)(276,438)(277,439)(278,437)(279,435)(280,445)(281,443)(282,450)(283,442)(284,449)(285,447)(286,448)(287,446)(288,444);;
s3 := (  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);;
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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s3*s2*s3*s2*s3*s2*s3, s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) 
s0 := 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)( 73,289)( 74,294)( 75,296)( 76,292)( 77,297)( 78,290)( 79,295)( 80,291)( 81,293)( 82,298)( 83,303)( 84,305)( 85,301)( 86,306)( 87,299)( 88,304)( 89,300)( 90,302)( 91,307)( 92,312)( 93,314)( 94,310)( 95,315)( 96,308)( 97,313)( 98,309)( 99,311)(100,316)(101,321)(102,323)(103,319)(104,324)(105,317)(106,322)(107,318)(108,320)(109,325)(110,330)(111,332)(112,328)(113,333)(114,326)(115,331)(116,327)(117,329)(118,334)(119,339)(120,341)(121,337)(122,342)(123,335)(124,340)(125,336)(126,338)(127,343)(128,348)(129,350)(130,346)(131,351)(132,344)(133,349)(134,345)(135,347)(136,352)(137,357)(138,359)(139,355)(140,360)(141,353)(142,358)(143,354)(144,356)(145,514)(146,519)(147,521)(148,517)(149,522)(150,515)(151,520)(152,516)(153,518)(154,505)(155,510)(156,512)(157,508)(158,513)(159,506)(160,511)(161,507)(162,509)(163,532)(164,537)(165,539)(166,535)(167,540)(168,533)(169,538)(170,534)(171,536)(172,523)(173,528)(174,530)(175,526)(176,531)(177,524)(178,529)(179,525)(180,527)(181,550)(182,555)(183,557)(184,553)(185,558)(186,551)(187,556)(188,552)(189,554)(190,541)(191,546)(192,548)(193,544)(194,549)(195,542)(196,547)(197,543)(198,545)(199,568)(200,573)(201,575)(202,571)(203,576)(204,569)(205,574)(206,570)(207,572)(208,559)(209,564)(210,566)(211,562)(212,567)(213,560)(214,565)(215,561)(216,563)(217,442)(218,447)(219,449)(220,445)(221,450)(222,443)(223,448)(224,444)(225,446)(226,433)(227,438)(228,440)(229,436)(230,441)(231,434)(232,439)(233,435)(234,437)(235,460)(236,465)(237,467)(238,463)(239,468)(240,461)(241,466)(242,462)(243,464)(244,451)(245,456)(246,458)(247,454)(248,459)(249,452)(250,457)(251,453)(252,455)(253,478)(254,483)(255,485)(256,481)(257,486)(258,479)(259,484)(260,480)(261,482)(262,469)(263,474)(264,476)(265,472)(266,477)(267,470)(268,475)(269,471)(270,473)(271,496)(272,501)(273,503)(274,499)(275,504)(276,497)(277,502)(278,498)(279,500)(280,487)(281,492)(282,494)(283,490)(284,495)(285,488)(286,493)(287,489)(288,491);
s1 := 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);
s2 := 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)( 73,337)( 74,335)( 75,342)( 76,334)( 77,341)( 78,339)( 79,340)( 80,338)( 81,336)( 82,328)( 83,326)( 84,333)( 85,325)( 86,332)( 87,330)( 88,331)( 89,329)( 90,327)( 91,355)( 92,353)( 93,360)( 94,352)( 95,359)( 96,357)( 97,358)( 98,356)( 99,354)(100,346)(101,344)(102,351)(103,343)(104,350)(105,348)(106,349)(107,347)(108,345)(109,301)(110,299)(111,306)(112,298)(113,305)(114,303)(115,304)(116,302)(117,300)(118,292)(119,290)(120,297)(121,289)(122,296)(123,294)(124,295)(125,293)(126,291)(127,319)(128,317)(129,324)(130,316)(131,323)(132,321)(133,322)(134,320)(135,318)(136,310)(137,308)(138,315)(139,307)(140,314)(141,312)(142,313)(143,311)(144,309)(145,526)(146,524)(147,531)(148,523)(149,530)(150,528)(151,529)(152,527)(153,525)(154,535)(155,533)(156,540)(157,532)(158,539)(159,537)(160,538)(161,536)(162,534)(163,508)(164,506)(165,513)(166,505)(167,512)(168,510)(169,511)(170,509)(171,507)(172,517)(173,515)(174,522)(175,514)(176,521)(177,519)(178,520)(179,518)(180,516)(181,562)(182,560)(183,567)(184,559)(185,566)(186,564)(187,565)(188,563)(189,561)(190,571)(191,569)(192,576)(193,568)(194,575)(195,573)(196,574)(197,572)(198,570)(199,544)(200,542)(201,549)(202,541)(203,548)(204,546)(205,547)(206,545)(207,543)(208,553)(209,551)(210,558)(211,550)(212,557)(213,555)(214,556)(215,554)(216,552)(217,490)(218,488)(219,495)(220,487)(221,494)(222,492)(223,493)(224,491)(225,489)(226,499)(227,497)(228,504)(229,496)(230,503)(231,501)(232,502)(233,500)(234,498)(235,472)(236,470)(237,477)(238,469)(239,476)(240,474)(241,475)(242,473)(243,471)(244,481)(245,479)(246,486)(247,478)(248,485)(249,483)(250,484)(251,482)(252,480)(253,454)(254,452)(255,459)(256,451)(257,458)(258,456)(259,457)(260,455)(261,453)(262,463)(263,461)(264,468)(265,460)(266,467)(267,465)(268,466)(269,464)(270,462)(271,436)(272,434)(273,441)(274,433)(275,440)(276,438)(277,439)(278,437)(279,435)(280,445)(281,443)(282,450)(283,442)(284,449)(285,447)(286,448)(287,446)(288,444);
s3 := 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);
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s3*s2*s3*s2*s3*s2*s3, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1 >;

References

  1. Theorem 10C12, McMullen P., Schulte, E.; Abstract Regular Polytopes (Cambridge University Press, 2002)

to this polytope.