Polytope of Type {8,4,22}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,4,22}*1408b
if this polytope has a name.
Group : SmallGroup(1408,13831)
Rank : 4
Schlafli Type : {8,4,22}
Number of vertices, edges, etc : 8, 16, 44, 22
Order of s₀s₁s₂s₃ : 88
Order of s₀s₁s₂s₃s₂s₁ : 2
Special Properties :
   Universal
   Orientable
   Flat
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,4,22}*704
   4-fold quotients : {2,4,22}*352, {4,2,22}*352
   8-fold quotients : {4,2,11}*176, {2,2,22}*176
   11-fold quotients : {8,4,2}*128b
   16-fold quotients : {2,2,11}*88
   22-fold quotients : {4,4,2}*64
   44-fold quotients : {2,4,2}*32, {4,2,2}*32
   88-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

s0 := (  1,353)(  2,354)(  3,355)(  4,356)(  5,357)(  6,358)(  7,359)(  8,360)
(  9,361)( 10,362)( 11,363)( 12,364)( 13,365)( 14,366)( 15,367)( 16,368)
( 17,369)( 18,370)( 19,371)( 20,372)( 21,373)( 22,374)( 23,386)( 24,387)
( 25,388)( 26,389)( 27,390)( 28,391)( 29,392)( 30,393)( 31,394)( 32,395)
( 33,396)( 34,375)( 35,376)( 36,377)( 37,378)( 38,379)( 39,380)( 40,381)
( 41,382)( 42,383)( 43,384)( 44,385)( 45,408)( 46,409)( 47,410)( 48,411)
( 49,412)( 50,413)( 51,414)( 52,415)( 53,416)( 54,417)( 55,418)( 56,397)
( 57,398)( 58,399)( 59,400)( 60,401)( 61,402)( 62,403)( 63,404)( 64,405)
( 65,406)( 66,407)( 67,419)( 68,420)( 69,421)( 70,422)( 71,423)( 72,424)
( 73,425)( 74,426)( 75,427)( 76,428)( 77,429)( 78,430)( 79,431)( 80,432)
( 81,433)( 82,434)( 83,435)( 84,436)( 85,437)( 86,438)( 87,439)( 88,440)
( 89,441)( 90,442)( 91,443)( 92,444)( 93,445)( 94,446)( 95,447)( 96,448)
( 97,449)( 98,450)( 99,451)(100,452)(101,453)(102,454)(103,455)(104,456)
(105,457)(106,458)(107,459)(108,460)(109,461)(110,462)(111,474)(112,475)
(113,476)(114,477)(115,478)(116,479)(117,480)(118,481)(119,482)(120,483)
(121,484)(122,463)(123,464)(124,465)(125,466)(126,467)(127,468)(128,469)
(129,470)(130,471)(131,472)(132,473)(133,496)(134,497)(135,498)(136,499)
(137,500)(138,501)(139,502)(140,503)(141,504)(142,505)(143,506)(144,485)
(145,486)(146,487)(147,488)(148,489)(149,490)(150,491)(151,492)(152,493)
(153,494)(154,495)(155,507)(156,508)(157,509)(158,510)(159,511)(160,512)
(161,513)(162,514)(163,515)(164,516)(165,517)(166,518)(167,519)(168,520)
(169,521)(170,522)(171,523)(172,524)(173,525)(174,526)(175,527)(176,528)
(177,529)(178,530)(179,531)(180,532)(181,533)(182,534)(183,535)(184,536)
(185,537)(186,538)(187,539)(188,540)(189,541)(190,542)(191,543)(192,544)
(193,545)(194,546)(195,547)(196,548)(197,549)(198,550)(199,562)(200,563)
(201,564)(202,565)(203,566)(204,567)(205,568)(206,569)(207,570)(208,571)
(209,572)(210,551)(211,552)(212,553)(213,554)(214,555)(215,556)(216,557)
(217,558)(218,559)(219,560)(220,561)(221,584)(222,585)(223,586)(224,587)
(225,588)(226,589)(227,590)(228,591)(229,592)(230,593)(231,594)(232,573)
(233,574)(234,575)(235,576)(236,577)(237,578)(238,579)(239,580)(240,581)
(241,582)(242,583)(243,595)(244,596)(245,597)(246,598)(247,599)(248,600)
(249,601)(250,602)(251,603)(252,604)(253,605)(254,606)(255,607)(256,608)
(257,609)(258,610)(259,611)(260,612)(261,613)(262,614)(263,615)(264,616)
(265,617)(266,618)(267,619)(268,620)(269,621)(270,622)(271,623)(272,624)
(273,625)(274,626)(275,627)(276,628)(277,629)(278,630)(279,631)(280,632)
(281,633)(282,634)(283,635)(284,636)(285,637)(286,638)(287,650)(288,651)
(289,652)(290,653)(291,654)(292,655)(293,656)(294,657)(295,658)(296,659)
(297,660)(298,639)(299,640)(300,641)(301,642)(302,643)(303,644)(304,645)
(305,646)(306,647)(307,648)(308,649)(309,672)(310,673)(311,674)(312,675)
(313,676)(314,677)(315,678)(316,679)(317,680)(318,681)(319,682)(320,661)
(321,662)(322,663)(323,664)(324,665)(325,666)(326,667)(327,668)(328,669)
(329,670)(330,671)(331,683)(332,684)(333,685)(334,686)(335,687)(336,688)
(337,689)(338,690)(339,691)(340,692)(341,693)(342,694)(343,695)(344,696)
(345,697)(346,698)(347,699)(348,700)(349,701)(350,702)(351,703)(352,704);;
s1 := ( 45, 56)( 46, 57)( 47, 58)( 48, 59)( 49, 60)( 50, 61)( 51, 62)( 52, 63)
( 53, 64)( 54, 65)( 55, 66)( 67, 78)( 68, 79)( 69, 80)( 70, 81)( 71, 82)
( 72, 83)( 73, 84)( 74, 85)( 75, 86)( 76, 87)( 77, 88)(133,144)(134,145)
(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)(142,153)
(143,154)(155,166)(156,167)(157,168)(158,169)(159,170)(160,171)(161,172)
(162,173)(163,174)(164,175)(165,176)(177,199)(178,200)(179,201)(180,202)
(181,203)(182,204)(183,205)(184,206)(185,207)(186,208)(187,209)(188,210)
(189,211)(190,212)(191,213)(192,214)(193,215)(194,216)(195,217)(196,218)
(197,219)(198,220)(221,254)(222,255)(223,256)(224,257)(225,258)(226,259)
(227,260)(228,261)(229,262)(230,263)(231,264)(232,243)(233,244)(234,245)
(235,246)(236,247)(237,248)(238,249)(239,250)(240,251)(241,252)(242,253)
(265,287)(266,288)(267,289)(268,290)(269,291)(270,292)(271,293)(272,294)
(273,295)(274,296)(275,297)(276,298)(277,299)(278,300)(279,301)(280,302)
(281,303)(282,304)(283,305)(284,306)(285,307)(286,308)(309,342)(310,343)
(311,344)(312,345)(313,346)(314,347)(315,348)(316,349)(317,350)(318,351)
(319,352)(320,331)(321,332)(322,333)(323,334)(324,335)(325,336)(326,337)
(327,338)(328,339)(329,340)(330,341)(353,397)(354,398)(355,399)(356,400)
(357,401)(358,402)(359,403)(360,404)(361,405)(362,406)(363,407)(364,408)
(365,409)(366,410)(367,411)(368,412)(369,413)(370,414)(371,415)(372,416)
(373,417)(374,418)(375,419)(376,420)(377,421)(378,422)(379,423)(380,424)
(381,425)(382,426)(383,427)(384,428)(385,429)(386,430)(387,431)(388,432)
(389,433)(390,434)(391,435)(392,436)(393,437)(394,438)(395,439)(396,440)
(441,485)(442,486)(443,487)(444,488)(445,489)(446,490)(447,491)(448,492)
(449,493)(450,494)(451,495)(452,496)(453,497)(454,498)(455,499)(456,500)
(457,501)(458,502)(459,503)(460,504)(461,505)(462,506)(463,507)(464,508)
(465,509)(466,510)(467,511)(468,512)(469,513)(470,514)(471,515)(472,516)
(473,517)(474,518)(475,519)(476,520)(477,521)(478,522)(479,523)(480,524)
(481,525)(482,526)(483,527)(484,528)(529,606)(530,607)(531,608)(532,609)
(533,610)(534,611)(535,612)(536,613)(537,614)(538,615)(539,616)(540,595)
(541,596)(542,597)(543,598)(544,599)(545,600)(546,601)(547,602)(548,603)
(549,604)(550,605)(551,584)(552,585)(553,586)(554,587)(555,588)(556,589)
(557,590)(558,591)(559,592)(560,593)(561,594)(562,573)(563,574)(564,575)
(565,576)(566,577)(567,578)(568,579)(569,580)(570,581)(571,582)(572,583)
(617,694)(618,695)(619,696)(620,697)(621,698)(622,699)(623,700)(624,701)
(625,702)(626,703)(627,704)(628,683)(629,684)(630,685)(631,686)(632,687)
(633,688)(634,689)(635,690)(636,691)(637,692)(638,693)(639,672)(640,673)
(641,674)(642,675)(643,676)(644,677)(645,678)(646,679)(647,680)(648,681)
(649,682)(650,661)(651,662)(652,663)(653,664)(654,665)(655,666)(656,667)
(657,668)(658,669)(659,670)(660,671);;
s2 := (  1,265)(  2,275)(  3,274)(  4,273)(  5,272)(  6,271)(  7,270)(  8,269)
(  9,268)( 10,267)( 11,266)( 12,276)( 13,286)( 14,285)( 15,284)( 16,283)
( 17,282)( 18,281)( 19,280)( 20,279)( 21,278)( 22,277)( 23,287)( 24,297)
( 25,296)( 26,295)( 27,294)( 28,293)( 29,292)( 30,291)( 31,290)( 32,289)
( 33,288)( 34,298)( 35,308)( 36,307)( 37,306)( 38,305)( 39,304)( 40,303)
( 41,302)( 42,301)( 43,300)( 44,299)( 45,320)( 46,330)( 47,329)( 48,328)
( 49,327)( 50,326)( 51,325)( 52,324)( 53,323)( 54,322)( 55,321)( 56,309)
( 57,319)( 58,318)( 59,317)( 60,316)( 61,315)( 62,314)( 63,313)( 64,312)
( 65,311)( 66,310)( 67,342)( 68,352)( 69,351)( 70,350)( 71,349)( 72,348)
( 73,347)( 74,346)( 75,345)( 76,344)( 77,343)( 78,331)( 79,341)( 80,340)
( 81,339)( 82,338)( 83,337)( 84,336)( 85,335)( 86,334)( 87,333)( 88,332)
( 89,177)( 90,187)( 91,186)( 92,185)( 93,184)( 94,183)( 95,182)( 96,181)
( 97,180)( 98,179)( 99,178)(100,188)(101,198)(102,197)(103,196)(104,195)
(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,199)(112,209)
(113,208)(114,207)(115,206)(116,205)(117,204)(118,203)(119,202)(120,201)
(121,200)(122,210)(123,220)(124,219)(125,218)(126,217)(127,216)(128,215)
(129,214)(130,213)(131,212)(132,211)(133,232)(134,242)(135,241)(136,240)
(137,239)(138,238)(139,237)(140,236)(141,235)(142,234)(143,233)(144,221)
(145,231)(146,230)(147,229)(148,228)(149,227)(150,226)(151,225)(152,224)
(153,223)(154,222)(155,254)(156,264)(157,263)(158,262)(159,261)(160,260)
(161,259)(162,258)(163,257)(164,256)(165,255)(166,243)(167,253)(168,252)
(169,251)(170,250)(171,249)(172,248)(173,247)(174,246)(175,245)(176,244)
(353,617)(354,627)(355,626)(356,625)(357,624)(358,623)(359,622)(360,621)
(361,620)(362,619)(363,618)(364,628)(365,638)(366,637)(367,636)(368,635)
(369,634)(370,633)(371,632)(372,631)(373,630)(374,629)(375,639)(376,649)
(377,648)(378,647)(379,646)(380,645)(381,644)(382,643)(383,642)(384,641)
(385,640)(386,650)(387,660)(388,659)(389,658)(390,657)(391,656)(392,655)
(393,654)(394,653)(395,652)(396,651)(397,672)(398,682)(399,681)(400,680)
(401,679)(402,678)(403,677)(404,676)(405,675)(406,674)(407,673)(408,661)
(409,671)(410,670)(411,669)(412,668)(413,667)(414,666)(415,665)(416,664)
(417,663)(418,662)(419,694)(420,704)(421,703)(422,702)(423,701)(424,700)
(425,699)(426,698)(427,697)(428,696)(429,695)(430,683)(431,693)(432,692)
(433,691)(434,690)(435,689)(436,688)(437,687)(438,686)(439,685)(440,684)
(441,529)(442,539)(443,538)(444,537)(445,536)(446,535)(447,534)(448,533)
(449,532)(450,531)(451,530)(452,540)(453,550)(454,549)(455,548)(456,547)
(457,546)(458,545)(459,544)(460,543)(461,542)(462,541)(463,551)(464,561)
(465,560)(466,559)(467,558)(468,557)(469,556)(470,555)(471,554)(472,553)
(473,552)(474,562)(475,572)(476,571)(477,570)(478,569)(479,568)(480,567)
(481,566)(482,565)(483,564)(484,563)(485,584)(486,594)(487,593)(488,592)
(489,591)(490,590)(491,589)(492,588)(493,587)(494,586)(495,585)(496,573)
(497,583)(498,582)(499,581)(500,580)(501,579)(502,578)(503,577)(504,576)
(505,575)(506,574)(507,606)(508,616)(509,615)(510,614)(511,613)(512,612)
(513,611)(514,610)(515,609)(516,608)(517,607)(518,595)(519,605)(520,604)
(521,603)(522,602)(523,601)(524,600)(525,599)(526,598)(527,597)(528,596);;
s3 := (  1, 90)(  2, 89)(  3, 99)(  4, 98)(  5, 97)(  6, 96)(  7, 95)(  8, 94)
(  9, 93)( 10, 92)( 11, 91)( 12,101)( 13,100)( 14,110)( 15,109)( 16,108)
( 17,107)( 18,106)( 19,105)( 20,104)( 21,103)( 22,102)( 23,112)( 24,111)
( 25,121)( 26,120)( 27,119)( 28,118)( 29,117)( 30,116)( 31,115)( 32,114)
( 33,113)( 34,123)( 35,122)( 36,132)( 37,131)( 38,130)( 39,129)( 40,128)
( 41,127)( 42,126)( 43,125)( 44,124)( 45,134)( 46,133)( 47,143)( 48,142)
( 49,141)( 50,140)( 51,139)( 52,138)( 53,137)( 54,136)( 55,135)( 56,145)
( 57,144)( 58,154)( 59,153)( 60,152)( 61,151)( 62,150)( 63,149)( 64,148)
( 65,147)( 66,146)( 67,156)( 68,155)( 69,165)( 70,164)( 71,163)( 72,162)
( 73,161)( 74,160)( 75,159)( 76,158)( 77,157)( 78,167)( 79,166)( 80,176)
( 81,175)( 82,174)( 83,173)( 84,172)( 85,171)( 86,170)( 87,169)( 88,168)
(177,266)(178,265)(179,275)(180,274)(181,273)(182,272)(183,271)(184,270)
(185,269)(186,268)(187,267)(188,277)(189,276)(190,286)(191,285)(192,284)
(193,283)(194,282)(195,281)(196,280)(197,279)(198,278)(199,288)(200,287)
(201,297)(202,296)(203,295)(204,294)(205,293)(206,292)(207,291)(208,290)
(209,289)(210,299)(211,298)(212,308)(213,307)(214,306)(215,305)(216,304)
(217,303)(218,302)(219,301)(220,300)(221,310)(222,309)(223,319)(224,318)
(225,317)(226,316)(227,315)(228,314)(229,313)(230,312)(231,311)(232,321)
(233,320)(234,330)(235,329)(236,328)(237,327)(238,326)(239,325)(240,324)
(241,323)(242,322)(243,332)(244,331)(245,341)(246,340)(247,339)(248,338)
(249,337)(250,336)(251,335)(252,334)(253,333)(254,343)(255,342)(256,352)
(257,351)(258,350)(259,349)(260,348)(261,347)(262,346)(263,345)(264,344)
(353,442)(354,441)(355,451)(356,450)(357,449)(358,448)(359,447)(360,446)
(361,445)(362,444)(363,443)(364,453)(365,452)(366,462)(367,461)(368,460)
(369,459)(370,458)(371,457)(372,456)(373,455)(374,454)(375,464)(376,463)
(377,473)(378,472)(379,471)(380,470)(381,469)(382,468)(383,467)(384,466)
(385,465)(386,475)(387,474)(388,484)(389,483)(390,482)(391,481)(392,480)
(393,479)(394,478)(395,477)(396,476)(397,486)(398,485)(399,495)(400,494)
(401,493)(402,492)(403,491)(404,490)(405,489)(406,488)(407,487)(408,497)
(409,496)(410,506)(411,505)(412,504)(413,503)(414,502)(415,501)(416,500)
(417,499)(418,498)(419,508)(420,507)(421,517)(422,516)(423,515)(424,514)
(425,513)(426,512)(427,511)(428,510)(429,509)(430,519)(431,518)(432,528)
(433,527)(434,526)(435,525)(436,524)(437,523)(438,522)(439,521)(440,520)
(529,618)(530,617)(531,627)(532,626)(533,625)(534,624)(535,623)(536,622)
(537,621)(538,620)(539,619)(540,629)(541,628)(542,638)(543,637)(544,636)
(545,635)(546,634)(547,633)(548,632)(549,631)(550,630)(551,640)(552,639)
(553,649)(554,648)(555,647)(556,646)(557,645)(558,644)(559,643)(560,642)
(561,641)(562,651)(563,650)(564,660)(565,659)(566,658)(567,657)(568,656)
(569,655)(570,654)(571,653)(572,652)(573,662)(574,661)(575,671)(576,670)
(577,669)(578,668)(579,667)(580,666)(581,665)(582,664)(583,663)(584,673)
(585,672)(586,682)(587,681)(588,680)(589,679)(590,678)(591,677)(592,676)
(593,675)(594,674)(595,684)(596,683)(597,693)(598,692)(599,691)(600,690)
(601,689)(602,688)(603,687)(604,686)(605,685)(606,695)(607,694)(608,704)
(609,703)(610,702)(611,701)(612,700)(613,699)(614,698)(615,697)(616,696);;
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, s1*s2*s1*s2*s1*s2*s1*s2, 
s1*s2*s3*s2*s1*s2*s3*s2, s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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(704)!(  1,353)(  2,354)(  3,355)(  4,356)(  5,357)(  6,358)(  7,359)
(  8,360)(  9,361)( 10,362)( 11,363)( 12,364)( 13,365)( 14,366)( 15,367)
( 16,368)( 17,369)( 18,370)( 19,371)( 20,372)( 21,373)( 22,374)( 23,386)
( 24,387)( 25,388)( 26,389)( 27,390)( 28,391)( 29,392)( 30,393)( 31,394)
( 32,395)( 33,396)( 34,375)( 35,376)( 36,377)( 37,378)( 38,379)( 39,380)
( 40,381)( 41,382)( 42,383)( 43,384)( 44,385)( 45,408)( 46,409)( 47,410)
( 48,411)( 49,412)( 50,413)( 51,414)( 52,415)( 53,416)( 54,417)( 55,418)
( 56,397)( 57,398)( 58,399)( 59,400)( 60,401)( 61,402)( 62,403)( 63,404)
( 64,405)( 65,406)( 66,407)( 67,419)( 68,420)( 69,421)( 70,422)( 71,423)
( 72,424)( 73,425)( 74,426)( 75,427)( 76,428)( 77,429)( 78,430)( 79,431)
( 80,432)( 81,433)( 82,434)( 83,435)( 84,436)( 85,437)( 86,438)( 87,439)
( 88,440)( 89,441)( 90,442)( 91,443)( 92,444)( 93,445)( 94,446)( 95,447)
( 96,448)( 97,449)( 98,450)( 99,451)(100,452)(101,453)(102,454)(103,455)
(104,456)(105,457)(106,458)(107,459)(108,460)(109,461)(110,462)(111,474)
(112,475)(113,476)(114,477)(115,478)(116,479)(117,480)(118,481)(119,482)
(120,483)(121,484)(122,463)(123,464)(124,465)(125,466)(126,467)(127,468)
(128,469)(129,470)(130,471)(131,472)(132,473)(133,496)(134,497)(135,498)
(136,499)(137,500)(138,501)(139,502)(140,503)(141,504)(142,505)(143,506)
(144,485)(145,486)(146,487)(147,488)(148,489)(149,490)(150,491)(151,492)
(152,493)(153,494)(154,495)(155,507)(156,508)(157,509)(158,510)(159,511)
(160,512)(161,513)(162,514)(163,515)(164,516)(165,517)(166,518)(167,519)
(168,520)(169,521)(170,522)(171,523)(172,524)(173,525)(174,526)(175,527)
(176,528)(177,529)(178,530)(179,531)(180,532)(181,533)(182,534)(183,535)
(184,536)(185,537)(186,538)(187,539)(188,540)(189,541)(190,542)(191,543)
(192,544)(193,545)(194,546)(195,547)(196,548)(197,549)(198,550)(199,562)
(200,563)(201,564)(202,565)(203,566)(204,567)(205,568)(206,569)(207,570)
(208,571)(209,572)(210,551)(211,552)(212,553)(213,554)(214,555)(215,556)
(216,557)(217,558)(218,559)(219,560)(220,561)(221,584)(222,585)(223,586)
(224,587)(225,588)(226,589)(227,590)(228,591)(229,592)(230,593)(231,594)
(232,573)(233,574)(234,575)(235,576)(236,577)(237,578)(238,579)(239,580)
(240,581)(241,582)(242,583)(243,595)(244,596)(245,597)(246,598)(247,599)
(248,600)(249,601)(250,602)(251,603)(252,604)(253,605)(254,606)(255,607)
(256,608)(257,609)(258,610)(259,611)(260,612)(261,613)(262,614)(263,615)
(264,616)(265,617)(266,618)(267,619)(268,620)(269,621)(270,622)(271,623)
(272,624)(273,625)(274,626)(275,627)(276,628)(277,629)(278,630)(279,631)
(280,632)(281,633)(282,634)(283,635)(284,636)(285,637)(286,638)(287,650)
(288,651)(289,652)(290,653)(291,654)(292,655)(293,656)(294,657)(295,658)
(296,659)(297,660)(298,639)(299,640)(300,641)(301,642)(302,643)(303,644)
(304,645)(305,646)(306,647)(307,648)(308,649)(309,672)(310,673)(311,674)
(312,675)(313,676)(314,677)(315,678)(316,679)(317,680)(318,681)(319,682)
(320,661)(321,662)(322,663)(323,664)(324,665)(325,666)(326,667)(327,668)
(328,669)(329,670)(330,671)(331,683)(332,684)(333,685)(334,686)(335,687)
(336,688)(337,689)(338,690)(339,691)(340,692)(341,693)(342,694)(343,695)
(344,696)(345,697)(346,698)(347,699)(348,700)(349,701)(350,702)(351,703)
(352,704);
s1 := Sym(704)!( 45, 56)( 46, 57)( 47, 58)( 48, 59)( 49, 60)( 50, 61)( 51, 62)
( 52, 63)( 53, 64)( 54, 65)( 55, 66)( 67, 78)( 68, 79)( 69, 80)( 70, 81)
( 71, 82)( 72, 83)( 73, 84)( 74, 85)( 75, 86)( 76, 87)( 77, 88)(133,144)
(134,145)(135,146)(136,147)(137,148)(138,149)(139,150)(140,151)(141,152)
(142,153)(143,154)(155,166)(156,167)(157,168)(158,169)(159,170)(160,171)
(161,172)(162,173)(163,174)(164,175)(165,176)(177,199)(178,200)(179,201)
(180,202)(181,203)(182,204)(183,205)(184,206)(185,207)(186,208)(187,209)
(188,210)(189,211)(190,212)(191,213)(192,214)(193,215)(194,216)(195,217)
(196,218)(197,219)(198,220)(221,254)(222,255)(223,256)(224,257)(225,258)
(226,259)(227,260)(228,261)(229,262)(230,263)(231,264)(232,243)(233,244)
(234,245)(235,246)(236,247)(237,248)(238,249)(239,250)(240,251)(241,252)
(242,253)(265,287)(266,288)(267,289)(268,290)(269,291)(270,292)(271,293)
(272,294)(273,295)(274,296)(275,297)(276,298)(277,299)(278,300)(279,301)
(280,302)(281,303)(282,304)(283,305)(284,306)(285,307)(286,308)(309,342)
(310,343)(311,344)(312,345)(313,346)(314,347)(315,348)(316,349)(317,350)
(318,351)(319,352)(320,331)(321,332)(322,333)(323,334)(324,335)(325,336)
(326,337)(327,338)(328,339)(329,340)(330,341)(353,397)(354,398)(355,399)
(356,400)(357,401)(358,402)(359,403)(360,404)(361,405)(362,406)(363,407)
(364,408)(365,409)(366,410)(367,411)(368,412)(369,413)(370,414)(371,415)
(372,416)(373,417)(374,418)(375,419)(376,420)(377,421)(378,422)(379,423)
(380,424)(381,425)(382,426)(383,427)(384,428)(385,429)(386,430)(387,431)
(388,432)(389,433)(390,434)(391,435)(392,436)(393,437)(394,438)(395,439)
(396,440)(441,485)(442,486)(443,487)(444,488)(445,489)(446,490)(447,491)
(448,492)(449,493)(450,494)(451,495)(452,496)(453,497)(454,498)(455,499)
(456,500)(457,501)(458,502)(459,503)(460,504)(461,505)(462,506)(463,507)
(464,508)(465,509)(466,510)(467,511)(468,512)(469,513)(470,514)(471,515)
(472,516)(473,517)(474,518)(475,519)(476,520)(477,521)(478,522)(479,523)
(480,524)(481,525)(482,526)(483,527)(484,528)(529,606)(530,607)(531,608)
(532,609)(533,610)(534,611)(535,612)(536,613)(537,614)(538,615)(539,616)
(540,595)(541,596)(542,597)(543,598)(544,599)(545,600)(546,601)(547,602)
(548,603)(549,604)(550,605)(551,584)(552,585)(553,586)(554,587)(555,588)
(556,589)(557,590)(558,591)(559,592)(560,593)(561,594)(562,573)(563,574)
(564,575)(565,576)(566,577)(567,578)(568,579)(569,580)(570,581)(571,582)
(572,583)(617,694)(618,695)(619,696)(620,697)(621,698)(622,699)(623,700)
(624,701)(625,702)(626,703)(627,704)(628,683)(629,684)(630,685)(631,686)
(632,687)(633,688)(634,689)(635,690)(636,691)(637,692)(638,693)(639,672)
(640,673)(641,674)(642,675)(643,676)(644,677)(645,678)(646,679)(647,680)
(648,681)(649,682)(650,661)(651,662)(652,663)(653,664)(654,665)(655,666)
(656,667)(657,668)(658,669)(659,670)(660,671);
s2 := Sym(704)!(  1,265)(  2,275)(  3,274)(  4,273)(  5,272)(  6,271)(  7,270)
(  8,269)(  9,268)( 10,267)( 11,266)( 12,276)( 13,286)( 14,285)( 15,284)
( 16,283)( 17,282)( 18,281)( 19,280)( 20,279)( 21,278)( 22,277)( 23,287)
( 24,297)( 25,296)( 26,295)( 27,294)( 28,293)( 29,292)( 30,291)( 31,290)
( 32,289)( 33,288)( 34,298)( 35,308)( 36,307)( 37,306)( 38,305)( 39,304)
( 40,303)( 41,302)( 42,301)( 43,300)( 44,299)( 45,320)( 46,330)( 47,329)
( 48,328)( 49,327)( 50,326)( 51,325)( 52,324)( 53,323)( 54,322)( 55,321)
( 56,309)( 57,319)( 58,318)( 59,317)( 60,316)( 61,315)( 62,314)( 63,313)
( 64,312)( 65,311)( 66,310)( 67,342)( 68,352)( 69,351)( 70,350)( 71,349)
( 72,348)( 73,347)( 74,346)( 75,345)( 76,344)( 77,343)( 78,331)( 79,341)
( 80,340)( 81,339)( 82,338)( 83,337)( 84,336)( 85,335)( 86,334)( 87,333)
( 88,332)( 89,177)( 90,187)( 91,186)( 92,185)( 93,184)( 94,183)( 95,182)
( 96,181)( 97,180)( 98,179)( 99,178)(100,188)(101,198)(102,197)(103,196)
(104,195)(105,194)(106,193)(107,192)(108,191)(109,190)(110,189)(111,199)
(112,209)(113,208)(114,207)(115,206)(116,205)(117,204)(118,203)(119,202)
(120,201)(121,200)(122,210)(123,220)(124,219)(125,218)(126,217)(127,216)
(128,215)(129,214)(130,213)(131,212)(132,211)(133,232)(134,242)(135,241)
(136,240)(137,239)(138,238)(139,237)(140,236)(141,235)(142,234)(143,233)
(144,221)(145,231)(146,230)(147,229)(148,228)(149,227)(150,226)(151,225)
(152,224)(153,223)(154,222)(155,254)(156,264)(157,263)(158,262)(159,261)
(160,260)(161,259)(162,258)(163,257)(164,256)(165,255)(166,243)(167,253)
(168,252)(169,251)(170,250)(171,249)(172,248)(173,247)(174,246)(175,245)
(176,244)(353,617)(354,627)(355,626)(356,625)(357,624)(358,623)(359,622)
(360,621)(361,620)(362,619)(363,618)(364,628)(365,638)(366,637)(367,636)
(368,635)(369,634)(370,633)(371,632)(372,631)(373,630)(374,629)(375,639)
(376,649)(377,648)(378,647)(379,646)(380,645)(381,644)(382,643)(383,642)
(384,641)(385,640)(386,650)(387,660)(388,659)(389,658)(390,657)(391,656)
(392,655)(393,654)(394,653)(395,652)(396,651)(397,672)(398,682)(399,681)
(400,680)(401,679)(402,678)(403,677)(404,676)(405,675)(406,674)(407,673)
(408,661)(409,671)(410,670)(411,669)(412,668)(413,667)(414,666)(415,665)
(416,664)(417,663)(418,662)(419,694)(420,704)(421,703)(422,702)(423,701)
(424,700)(425,699)(426,698)(427,697)(428,696)(429,695)(430,683)(431,693)
(432,692)(433,691)(434,690)(435,689)(436,688)(437,687)(438,686)(439,685)
(440,684)(441,529)(442,539)(443,538)(444,537)(445,536)(446,535)(447,534)
(448,533)(449,532)(450,531)(451,530)(452,540)(453,550)(454,549)(455,548)
(456,547)(457,546)(458,545)(459,544)(460,543)(461,542)(462,541)(463,551)
(464,561)(465,560)(466,559)(467,558)(468,557)(469,556)(470,555)(471,554)
(472,553)(473,552)(474,562)(475,572)(476,571)(477,570)(478,569)(479,568)
(480,567)(481,566)(482,565)(483,564)(484,563)(485,584)(486,594)(487,593)
(488,592)(489,591)(490,590)(491,589)(492,588)(493,587)(494,586)(495,585)
(496,573)(497,583)(498,582)(499,581)(500,580)(501,579)(502,578)(503,577)
(504,576)(505,575)(506,574)(507,606)(508,616)(509,615)(510,614)(511,613)
(512,612)(513,611)(514,610)(515,609)(516,608)(517,607)(518,595)(519,605)
(520,604)(521,603)(522,602)(523,601)(524,600)(525,599)(526,598)(527,597)
(528,596);
s3 := Sym(704)!(  1, 90)(  2, 89)(  3, 99)(  4, 98)(  5, 97)(  6, 96)(  7, 95)
(  8, 94)(  9, 93)( 10, 92)( 11, 91)( 12,101)( 13,100)( 14,110)( 15,109)
( 16,108)( 17,107)( 18,106)( 19,105)( 20,104)( 21,103)( 22,102)( 23,112)
( 24,111)( 25,121)( 26,120)( 27,119)( 28,118)( 29,117)( 30,116)( 31,115)
( 32,114)( 33,113)( 34,123)( 35,122)( 36,132)( 37,131)( 38,130)( 39,129)
( 40,128)( 41,127)( 42,126)( 43,125)( 44,124)( 45,134)( 46,133)( 47,143)
( 48,142)( 49,141)( 50,140)( 51,139)( 52,138)( 53,137)( 54,136)( 55,135)
( 56,145)( 57,144)( 58,154)( 59,153)( 60,152)( 61,151)( 62,150)( 63,149)
( 64,148)( 65,147)( 66,146)( 67,156)( 68,155)( 69,165)( 70,164)( 71,163)
( 72,162)( 73,161)( 74,160)( 75,159)( 76,158)( 77,157)( 78,167)( 79,166)
( 80,176)( 81,175)( 82,174)( 83,173)( 84,172)( 85,171)( 86,170)( 87,169)
( 88,168)(177,266)(178,265)(179,275)(180,274)(181,273)(182,272)(183,271)
(184,270)(185,269)(186,268)(187,267)(188,277)(189,276)(190,286)(191,285)
(192,284)(193,283)(194,282)(195,281)(196,280)(197,279)(198,278)(199,288)
(200,287)(201,297)(202,296)(203,295)(204,294)(205,293)(206,292)(207,291)
(208,290)(209,289)(210,299)(211,298)(212,308)(213,307)(214,306)(215,305)
(216,304)(217,303)(218,302)(219,301)(220,300)(221,310)(222,309)(223,319)
(224,318)(225,317)(226,316)(227,315)(228,314)(229,313)(230,312)(231,311)
(232,321)(233,320)(234,330)(235,329)(236,328)(237,327)(238,326)(239,325)
(240,324)(241,323)(242,322)(243,332)(244,331)(245,341)(246,340)(247,339)
(248,338)(249,337)(250,336)(251,335)(252,334)(253,333)(254,343)(255,342)
(256,352)(257,351)(258,350)(259,349)(260,348)(261,347)(262,346)(263,345)
(264,344)(353,442)(354,441)(355,451)(356,450)(357,449)(358,448)(359,447)
(360,446)(361,445)(362,444)(363,443)(364,453)(365,452)(366,462)(367,461)
(368,460)(369,459)(370,458)(371,457)(372,456)(373,455)(374,454)(375,464)
(376,463)(377,473)(378,472)(379,471)(380,470)(381,469)(382,468)(383,467)
(384,466)(385,465)(386,475)(387,474)(388,484)(389,483)(390,482)(391,481)
(392,480)(393,479)(394,478)(395,477)(396,476)(397,486)(398,485)(399,495)
(400,494)(401,493)(402,492)(403,491)(404,490)(405,489)(406,488)(407,487)
(408,497)(409,496)(410,506)(411,505)(412,504)(413,503)(414,502)(415,501)
(416,500)(417,499)(418,498)(419,508)(420,507)(421,517)(422,516)(423,515)
(424,514)(425,513)(426,512)(427,511)(428,510)(429,509)(430,519)(431,518)
(432,528)(433,527)(434,526)(435,525)(436,524)(437,523)(438,522)(439,521)
(440,520)(529,618)(530,617)(531,627)(532,626)(533,625)(534,624)(535,623)
(536,622)(537,621)(538,620)(539,619)(540,629)(541,628)(542,638)(543,637)
(544,636)(545,635)(546,634)(547,633)(548,632)(549,631)(550,630)(551,640)
(552,639)(553,649)(554,648)(555,647)(556,646)(557,645)(558,644)(559,643)
(560,642)(561,641)(562,651)(563,650)(564,660)(565,659)(566,658)(567,657)
(568,656)(569,655)(570,654)(571,653)(572,652)(573,662)(574,661)(575,671)
(576,670)(577,669)(578,668)(579,667)(580,666)(581,665)(582,664)(583,663)
(584,673)(585,672)(586,682)(587,681)(588,680)(589,679)(590,678)(591,677)
(592,676)(593,675)(594,674)(595,684)(596,683)(597,693)(598,692)(599,691)
(600,690)(601,689)(602,688)(603,687)(604,686)(605,685)(606,695)(607,694)
(608,704)(609,703)(610,702)(611,701)(612,700)(613,699)(614,698)(615,697)
(616,696);
poly := sub<Sym(704)|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, 
s1*s2*s1*s2*s1*s2*s1*s2, s1*s2*s3*s2*s1*s2*s3*s2, 
s2*s0*s1*s0*s1*s2*s0*s1*s0*s1, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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