Polytope of Type {4,6,28}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,6,28}*1344a
Also Known As : {{4,6|2},{6,28|2}}. if this polytope has another name.
Group : SmallGroup(1344,7535)
Rank : 4
Schlafli Type : {4,6,28}
Number of vertices, edges, etc : 4, 12, 84, 28
Order of s0s1s2s3 : 84
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 : {2,6,28}*672a, {4,6,14}*672a
3-fold quotients : {4,2,28}*448
4-fold quotients : {2,6,14}*336
6-fold quotients : {2,2,28}*224, {4,2,14}*224
7-fold quotients : {4,6,4}*192a
12-fold quotients : {4,2,7}*112, {2,2,14}*112
14-fold quotients : {2,6,4}*96a, {4,6,2}*96a
21-fold quotients : {4,2,4}*64
24-fold quotients : {2,2,7}*56
28-fold quotients : {2,6,2}*48
42-fold quotients : {2,2,4}*32, {4,2,2}*32
56-fold quotients : {2,3,2}*24
84-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 1,169)( 2,170)( 3,171)( 4,172)( 5,173)( 6,174)( 7,175)( 8,176)
( 9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)( 16,184)
( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)( 24,192)
( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)( 32,200)
( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)( 40,208)
( 41,209)( 42,210)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)( 48,216)
( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)( 56,224)
( 57,225)( 58,226)( 59,227)( 60,228)( 61,229)( 62,230)( 63,231)( 64,232)
( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)( 72,240)
( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)( 80,248)
( 81,249)( 82,250)( 83,251)( 84,252)( 85,295)( 86,296)( 87,297)( 88,298)
( 89,299)( 90,300)( 91,301)( 92,302)( 93,303)( 94,304)( 95,305)( 96,306)
( 97,307)( 98,308)( 99,309)(100,310)(101,311)(102,312)(103,313)(104,314)
(105,315)(106,316)(107,317)(108,318)(109,319)(110,320)(111,321)(112,322)
(113,323)(114,324)(115,325)(116,326)(117,327)(118,328)(119,329)(120,330)
(121,331)(122,332)(123,333)(124,334)(125,335)(126,336)(127,253)(128,254)
(129,255)(130,256)(131,257)(132,258)(133,259)(134,260)(135,261)(136,262)
(137,263)(138,264)(139,265)(140,266)(141,267)(142,268)(143,269)(144,270)
(145,271)(146,272)(147,273)(148,274)(149,275)(150,276)(151,277)(152,278)
(153,279)(154,280)(155,281)(156,282)(157,283)(158,284)(159,285)(160,286)
(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)(168,294)
(337,505)(338,506)(339,507)(340,508)(341,509)(342,510)(343,511)(344,512)
(345,513)(346,514)(347,515)(348,516)(349,517)(350,518)(351,519)(352,520)
(353,521)(354,522)(355,523)(356,524)(357,525)(358,526)(359,527)(360,528)
(361,529)(362,530)(363,531)(364,532)(365,533)(366,534)(367,535)(368,536)
(369,537)(370,538)(371,539)(372,540)(373,541)(374,542)(375,543)(376,544)
(377,545)(378,546)(379,547)(380,548)(381,549)(382,550)(383,551)(384,552)
(385,553)(386,554)(387,555)(388,556)(389,557)(390,558)(391,559)(392,560)
(393,561)(394,562)(395,563)(396,564)(397,565)(398,566)(399,567)(400,568)
(401,569)(402,570)(403,571)(404,572)(405,573)(406,574)(407,575)(408,576)
(409,577)(410,578)(411,579)(412,580)(413,581)(414,582)(415,583)(416,584)
(417,585)(418,586)(419,587)(420,588)(421,631)(422,632)(423,633)(424,634)
(425,635)(426,636)(427,637)(428,638)(429,639)(430,640)(431,641)(432,642)
(433,643)(434,644)(435,645)(436,646)(437,647)(438,648)(439,649)(440,650)
(441,651)(442,652)(443,653)(444,654)(445,655)(446,656)(447,657)(448,658)
(449,659)(450,660)(451,661)(452,662)(453,663)(454,664)(455,665)(456,666)
(457,667)(458,668)(459,669)(460,670)(461,671)(462,672)(463,589)(464,590)
(465,591)(466,592)(467,593)(468,594)(469,595)(470,596)(471,597)(472,598)
(473,599)(474,600)(475,601)(476,602)(477,603)(478,604)(479,605)(480,606)
(481,607)(482,608)(483,609)(484,610)(485,611)(486,612)(487,613)(488,614)
(489,615)(490,616)(491,617)(492,618)(493,619)(494,620)(495,621)(496,622)
(497,623)(498,624)(499,625)(500,626)(501,627)(502,628)(503,629)(504,630);;
s1 := ( 1,421)( 2,422)( 3,423)( 4,424)( 5,425)( 6,426)( 7,427)( 8,435)
( 9,436)( 10,437)( 11,438)( 12,439)( 13,440)( 14,441)( 15,428)( 16,429)
( 17,430)( 18,431)( 19,432)( 20,433)( 21,434)( 22,442)( 23,443)( 24,444)
( 25,445)( 26,446)( 27,447)( 28,448)( 29,456)( 30,457)( 31,458)( 32,459)
( 33,460)( 34,461)( 35,462)( 36,449)( 37,450)( 38,451)( 39,452)( 40,453)
( 41,454)( 42,455)( 43,463)( 44,464)( 45,465)( 46,466)( 47,467)( 48,468)
( 49,469)( 50,477)( 51,478)( 52,479)( 53,480)( 54,481)( 55,482)( 56,483)
( 57,470)( 58,471)( 59,472)( 60,473)( 61,474)( 62,475)( 63,476)( 64,484)
( 65,485)( 66,486)( 67,487)( 68,488)( 69,489)( 70,490)( 71,498)( 72,499)
( 73,500)( 74,501)( 75,502)( 76,503)( 77,504)( 78,491)( 79,492)( 80,493)
( 81,494)( 82,495)( 83,496)( 84,497)( 85,358)( 86,359)( 87,360)( 88,361)
( 89,362)( 90,363)( 91,364)( 92,372)( 93,373)( 94,374)( 95,375)( 96,376)
( 97,377)( 98,378)( 99,365)(100,366)(101,367)(102,368)(103,369)(104,370)
(105,371)(106,337)(107,338)(108,339)(109,340)(110,341)(111,342)(112,343)
(113,351)(114,352)(115,353)(116,354)(117,355)(118,356)(119,357)(120,344)
(121,345)(122,346)(123,347)(124,348)(125,349)(126,350)(127,400)(128,401)
(129,402)(130,403)(131,404)(132,405)(133,406)(134,414)(135,415)(136,416)
(137,417)(138,418)(139,419)(140,420)(141,407)(142,408)(143,409)(144,410)
(145,411)(146,412)(147,413)(148,379)(149,380)(150,381)(151,382)(152,383)
(153,384)(154,385)(155,393)(156,394)(157,395)(158,396)(159,397)(160,398)
(161,399)(162,386)(163,387)(164,388)(165,389)(166,390)(167,391)(168,392)
(169,589)(170,590)(171,591)(172,592)(173,593)(174,594)(175,595)(176,603)
(177,604)(178,605)(179,606)(180,607)(181,608)(182,609)(183,596)(184,597)
(185,598)(186,599)(187,600)(188,601)(189,602)(190,610)(191,611)(192,612)
(193,613)(194,614)(195,615)(196,616)(197,624)(198,625)(199,626)(200,627)
(201,628)(202,629)(203,630)(204,617)(205,618)(206,619)(207,620)(208,621)
(209,622)(210,623)(211,631)(212,632)(213,633)(214,634)(215,635)(216,636)
(217,637)(218,645)(219,646)(220,647)(221,648)(222,649)(223,650)(224,651)
(225,638)(226,639)(227,640)(228,641)(229,642)(230,643)(231,644)(232,652)
(233,653)(234,654)(235,655)(236,656)(237,657)(238,658)(239,666)(240,667)
(241,668)(242,669)(243,670)(244,671)(245,672)(246,659)(247,660)(248,661)
(249,662)(250,663)(251,664)(252,665)(253,526)(254,527)(255,528)(256,529)
(257,530)(258,531)(259,532)(260,540)(261,541)(262,542)(263,543)(264,544)
(265,545)(266,546)(267,533)(268,534)(269,535)(270,536)(271,537)(272,538)
(273,539)(274,505)(275,506)(276,507)(277,508)(278,509)(279,510)(280,511)
(281,519)(282,520)(283,521)(284,522)(285,523)(286,524)(287,525)(288,512)
(289,513)(290,514)(291,515)(292,516)(293,517)(294,518)(295,568)(296,569)
(297,570)(298,571)(299,572)(300,573)(301,574)(302,582)(303,583)(304,584)
(305,585)(306,586)(307,587)(308,588)(309,575)(310,576)(311,577)(312,578)
(313,579)(314,580)(315,581)(316,547)(317,548)(318,549)(319,550)(320,551)
(321,552)(322,553)(323,561)(324,562)(325,563)(326,564)(327,565)(328,566)
(329,567)(330,554)(331,555)(332,556)(333,557)(334,558)(335,559)(336,560);;
s2 := ( 1,344)( 2,350)( 3,349)( 4,348)( 5,347)( 6,346)( 7,345)( 8,337)
( 9,343)( 10,342)( 11,341)( 12,340)( 13,339)( 14,338)( 15,351)( 16,357)
( 17,356)( 18,355)( 19,354)( 20,353)( 21,352)( 22,365)( 23,371)( 24,370)
( 25,369)( 26,368)( 27,367)( 28,366)( 29,358)( 30,364)( 31,363)( 32,362)
( 33,361)( 34,360)( 35,359)( 36,372)( 37,378)( 38,377)( 39,376)( 40,375)
( 41,374)( 42,373)( 43,386)( 44,392)( 45,391)( 46,390)( 47,389)( 48,388)
( 49,387)( 50,379)( 51,385)( 52,384)( 53,383)( 54,382)( 55,381)( 56,380)
( 57,393)( 58,399)( 59,398)( 60,397)( 61,396)( 62,395)( 63,394)( 64,407)
( 65,413)( 66,412)( 67,411)( 68,410)( 69,409)( 70,408)( 71,400)( 72,406)
( 73,405)( 74,404)( 75,403)( 76,402)( 77,401)( 78,414)( 79,420)( 80,419)
( 81,418)( 82,417)( 83,416)( 84,415)( 85,449)( 86,455)( 87,454)( 88,453)
( 89,452)( 90,451)( 91,450)( 92,442)( 93,448)( 94,447)( 95,446)( 96,445)
( 97,444)( 98,443)( 99,456)(100,462)(101,461)(102,460)(103,459)(104,458)
(105,457)(106,428)(107,434)(108,433)(109,432)(110,431)(111,430)(112,429)
(113,421)(114,427)(115,426)(116,425)(117,424)(118,423)(119,422)(120,435)
(121,441)(122,440)(123,439)(124,438)(125,437)(126,436)(127,491)(128,497)
(129,496)(130,495)(131,494)(132,493)(133,492)(134,484)(135,490)(136,489)
(137,488)(138,487)(139,486)(140,485)(141,498)(142,504)(143,503)(144,502)
(145,501)(146,500)(147,499)(148,470)(149,476)(150,475)(151,474)(152,473)
(153,472)(154,471)(155,463)(156,469)(157,468)(158,467)(159,466)(160,465)
(161,464)(162,477)(163,483)(164,482)(165,481)(166,480)(167,479)(168,478)
(169,512)(170,518)(171,517)(172,516)(173,515)(174,514)(175,513)(176,505)
(177,511)(178,510)(179,509)(180,508)(181,507)(182,506)(183,519)(184,525)
(185,524)(186,523)(187,522)(188,521)(189,520)(190,533)(191,539)(192,538)
(193,537)(194,536)(195,535)(196,534)(197,526)(198,532)(199,531)(200,530)
(201,529)(202,528)(203,527)(204,540)(205,546)(206,545)(207,544)(208,543)
(209,542)(210,541)(211,554)(212,560)(213,559)(214,558)(215,557)(216,556)
(217,555)(218,547)(219,553)(220,552)(221,551)(222,550)(223,549)(224,548)
(225,561)(226,567)(227,566)(228,565)(229,564)(230,563)(231,562)(232,575)
(233,581)(234,580)(235,579)(236,578)(237,577)(238,576)(239,568)(240,574)
(241,573)(242,572)(243,571)(244,570)(245,569)(246,582)(247,588)(248,587)
(249,586)(250,585)(251,584)(252,583)(253,617)(254,623)(255,622)(256,621)
(257,620)(258,619)(259,618)(260,610)(261,616)(262,615)(263,614)(264,613)
(265,612)(266,611)(267,624)(268,630)(269,629)(270,628)(271,627)(272,626)
(273,625)(274,596)(275,602)(276,601)(277,600)(278,599)(279,598)(280,597)
(281,589)(282,595)(283,594)(284,593)(285,592)(286,591)(287,590)(288,603)
(289,609)(290,608)(291,607)(292,606)(293,605)(294,604)(295,659)(296,665)
(297,664)(298,663)(299,662)(300,661)(301,660)(302,652)(303,658)(304,657)
(305,656)(306,655)(307,654)(308,653)(309,666)(310,672)(311,671)(312,670)
(313,669)(314,668)(315,667)(316,638)(317,644)(318,643)(319,642)(320,641)
(321,640)(322,639)(323,631)(324,637)(325,636)(326,635)(327,634)(328,633)
(329,632)(330,645)(331,651)(332,650)(333,649)(334,648)(335,647)(336,646);;
s3 := ( 1, 2)( 3, 7)( 4, 6)( 8, 9)( 10, 14)( 11, 13)( 15, 16)( 17, 21)
( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)( 36, 37)
( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)( 53, 55)
( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)( 73, 77)
( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,107)( 86,106)( 87,112)( 88,111)
( 89,110)( 90,109)( 91,108)( 92,114)( 93,113)( 94,119)( 95,118)( 96,117)
( 97,116)( 98,115)( 99,121)(100,120)(101,126)(102,125)(103,124)(104,123)
(105,122)(127,149)(128,148)(129,154)(130,153)(131,152)(132,151)(133,150)
(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)(141,163)
(142,162)(143,168)(144,167)(145,166)(146,165)(147,164)(169,170)(171,175)
(172,174)(176,177)(178,182)(179,181)(183,184)(185,189)(186,188)(190,191)
(192,196)(193,195)(197,198)(199,203)(200,202)(204,205)(206,210)(207,209)
(211,212)(213,217)(214,216)(218,219)(220,224)(221,223)(225,226)(227,231)
(228,230)(232,233)(234,238)(235,237)(239,240)(241,245)(242,244)(246,247)
(248,252)(249,251)(253,275)(254,274)(255,280)(256,279)(257,278)(258,277)
(259,276)(260,282)(261,281)(262,287)(263,286)(264,285)(265,284)(266,283)
(267,289)(268,288)(269,294)(270,293)(271,292)(272,291)(273,290)(295,317)
(296,316)(297,322)(298,321)(299,320)(300,319)(301,318)(302,324)(303,323)
(304,329)(305,328)(306,327)(307,326)(308,325)(309,331)(310,330)(311,336)
(312,335)(313,334)(314,333)(315,332)(337,359)(338,358)(339,364)(340,363)
(341,362)(342,361)(343,360)(344,366)(345,365)(346,371)(347,370)(348,369)
(349,368)(350,367)(351,373)(352,372)(353,378)(354,377)(355,376)(356,375)
(357,374)(379,401)(380,400)(381,406)(382,405)(383,404)(384,403)(385,402)
(386,408)(387,407)(388,413)(389,412)(390,411)(391,410)(392,409)(393,415)
(394,414)(395,420)(396,419)(397,418)(398,417)(399,416)(421,422)(423,427)
(424,426)(428,429)(430,434)(431,433)(435,436)(437,441)(438,440)(442,443)
(444,448)(445,447)(449,450)(451,455)(452,454)(456,457)(458,462)(459,461)
(463,464)(465,469)(466,468)(470,471)(472,476)(473,475)(477,478)(479,483)
(480,482)(484,485)(486,490)(487,489)(491,492)(493,497)(494,496)(498,499)
(500,504)(501,503)(505,527)(506,526)(507,532)(508,531)(509,530)(510,529)
(511,528)(512,534)(513,533)(514,539)(515,538)(516,537)(517,536)(518,535)
(519,541)(520,540)(521,546)(522,545)(523,544)(524,543)(525,542)(547,569)
(548,568)(549,574)(550,573)(551,572)(552,571)(553,570)(554,576)(555,575)
(556,581)(557,580)(558,579)(559,578)(560,577)(561,583)(562,582)(563,588)
(564,587)(565,586)(566,585)(567,584)(589,590)(591,595)(592,594)(596,597)
(598,602)(599,601)(603,604)(605,609)(606,608)(610,611)(612,616)(613,615)
(617,618)(619,623)(620,622)(624,625)(626,630)(627,629)(631,632)(633,637)
(634,636)(638,639)(640,644)(641,643)(645,646)(647,651)(648,650)(652,653)
(654,658)(655,657)(659,660)(661,665)(662,664)(666,667)(668,672)(669,671);;
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*s3*s2*s1*s2*s3*s2,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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(672)!( 1,169)( 2,170)( 3,171)( 4,172)( 5,173)( 6,174)( 7,175)
( 8,176)( 9,177)( 10,178)( 11,179)( 12,180)( 13,181)( 14,182)( 15,183)
( 16,184)( 17,185)( 18,186)( 19,187)( 20,188)( 21,189)( 22,190)( 23,191)
( 24,192)( 25,193)( 26,194)( 27,195)( 28,196)( 29,197)( 30,198)( 31,199)
( 32,200)( 33,201)( 34,202)( 35,203)( 36,204)( 37,205)( 38,206)( 39,207)
( 40,208)( 41,209)( 42,210)( 43,211)( 44,212)( 45,213)( 46,214)( 47,215)
( 48,216)( 49,217)( 50,218)( 51,219)( 52,220)( 53,221)( 54,222)( 55,223)
( 56,224)( 57,225)( 58,226)( 59,227)( 60,228)( 61,229)( 62,230)( 63,231)
( 64,232)( 65,233)( 66,234)( 67,235)( 68,236)( 69,237)( 70,238)( 71,239)
( 72,240)( 73,241)( 74,242)( 75,243)( 76,244)( 77,245)( 78,246)( 79,247)
( 80,248)( 81,249)( 82,250)( 83,251)( 84,252)( 85,295)( 86,296)( 87,297)
( 88,298)( 89,299)( 90,300)( 91,301)( 92,302)( 93,303)( 94,304)( 95,305)
( 96,306)( 97,307)( 98,308)( 99,309)(100,310)(101,311)(102,312)(103,313)
(104,314)(105,315)(106,316)(107,317)(108,318)(109,319)(110,320)(111,321)
(112,322)(113,323)(114,324)(115,325)(116,326)(117,327)(118,328)(119,329)
(120,330)(121,331)(122,332)(123,333)(124,334)(125,335)(126,336)(127,253)
(128,254)(129,255)(130,256)(131,257)(132,258)(133,259)(134,260)(135,261)
(136,262)(137,263)(138,264)(139,265)(140,266)(141,267)(142,268)(143,269)
(144,270)(145,271)(146,272)(147,273)(148,274)(149,275)(150,276)(151,277)
(152,278)(153,279)(154,280)(155,281)(156,282)(157,283)(158,284)(159,285)
(160,286)(161,287)(162,288)(163,289)(164,290)(165,291)(166,292)(167,293)
(168,294)(337,505)(338,506)(339,507)(340,508)(341,509)(342,510)(343,511)
(344,512)(345,513)(346,514)(347,515)(348,516)(349,517)(350,518)(351,519)
(352,520)(353,521)(354,522)(355,523)(356,524)(357,525)(358,526)(359,527)
(360,528)(361,529)(362,530)(363,531)(364,532)(365,533)(366,534)(367,535)
(368,536)(369,537)(370,538)(371,539)(372,540)(373,541)(374,542)(375,543)
(376,544)(377,545)(378,546)(379,547)(380,548)(381,549)(382,550)(383,551)
(384,552)(385,553)(386,554)(387,555)(388,556)(389,557)(390,558)(391,559)
(392,560)(393,561)(394,562)(395,563)(396,564)(397,565)(398,566)(399,567)
(400,568)(401,569)(402,570)(403,571)(404,572)(405,573)(406,574)(407,575)
(408,576)(409,577)(410,578)(411,579)(412,580)(413,581)(414,582)(415,583)
(416,584)(417,585)(418,586)(419,587)(420,588)(421,631)(422,632)(423,633)
(424,634)(425,635)(426,636)(427,637)(428,638)(429,639)(430,640)(431,641)
(432,642)(433,643)(434,644)(435,645)(436,646)(437,647)(438,648)(439,649)
(440,650)(441,651)(442,652)(443,653)(444,654)(445,655)(446,656)(447,657)
(448,658)(449,659)(450,660)(451,661)(452,662)(453,663)(454,664)(455,665)
(456,666)(457,667)(458,668)(459,669)(460,670)(461,671)(462,672)(463,589)
(464,590)(465,591)(466,592)(467,593)(468,594)(469,595)(470,596)(471,597)
(472,598)(473,599)(474,600)(475,601)(476,602)(477,603)(478,604)(479,605)
(480,606)(481,607)(482,608)(483,609)(484,610)(485,611)(486,612)(487,613)
(488,614)(489,615)(490,616)(491,617)(492,618)(493,619)(494,620)(495,621)
(496,622)(497,623)(498,624)(499,625)(500,626)(501,627)(502,628)(503,629)
(504,630);
s1 := Sym(672)!( 1,421)( 2,422)( 3,423)( 4,424)( 5,425)( 6,426)( 7,427)
( 8,435)( 9,436)( 10,437)( 11,438)( 12,439)( 13,440)( 14,441)( 15,428)
( 16,429)( 17,430)( 18,431)( 19,432)( 20,433)( 21,434)( 22,442)( 23,443)
( 24,444)( 25,445)( 26,446)( 27,447)( 28,448)( 29,456)( 30,457)( 31,458)
( 32,459)( 33,460)( 34,461)( 35,462)( 36,449)( 37,450)( 38,451)( 39,452)
( 40,453)( 41,454)( 42,455)( 43,463)( 44,464)( 45,465)( 46,466)( 47,467)
( 48,468)( 49,469)( 50,477)( 51,478)( 52,479)( 53,480)( 54,481)( 55,482)
( 56,483)( 57,470)( 58,471)( 59,472)( 60,473)( 61,474)( 62,475)( 63,476)
( 64,484)( 65,485)( 66,486)( 67,487)( 68,488)( 69,489)( 70,490)( 71,498)
( 72,499)( 73,500)( 74,501)( 75,502)( 76,503)( 77,504)( 78,491)( 79,492)
( 80,493)( 81,494)( 82,495)( 83,496)( 84,497)( 85,358)( 86,359)( 87,360)
( 88,361)( 89,362)( 90,363)( 91,364)( 92,372)( 93,373)( 94,374)( 95,375)
( 96,376)( 97,377)( 98,378)( 99,365)(100,366)(101,367)(102,368)(103,369)
(104,370)(105,371)(106,337)(107,338)(108,339)(109,340)(110,341)(111,342)
(112,343)(113,351)(114,352)(115,353)(116,354)(117,355)(118,356)(119,357)
(120,344)(121,345)(122,346)(123,347)(124,348)(125,349)(126,350)(127,400)
(128,401)(129,402)(130,403)(131,404)(132,405)(133,406)(134,414)(135,415)
(136,416)(137,417)(138,418)(139,419)(140,420)(141,407)(142,408)(143,409)
(144,410)(145,411)(146,412)(147,413)(148,379)(149,380)(150,381)(151,382)
(152,383)(153,384)(154,385)(155,393)(156,394)(157,395)(158,396)(159,397)
(160,398)(161,399)(162,386)(163,387)(164,388)(165,389)(166,390)(167,391)
(168,392)(169,589)(170,590)(171,591)(172,592)(173,593)(174,594)(175,595)
(176,603)(177,604)(178,605)(179,606)(180,607)(181,608)(182,609)(183,596)
(184,597)(185,598)(186,599)(187,600)(188,601)(189,602)(190,610)(191,611)
(192,612)(193,613)(194,614)(195,615)(196,616)(197,624)(198,625)(199,626)
(200,627)(201,628)(202,629)(203,630)(204,617)(205,618)(206,619)(207,620)
(208,621)(209,622)(210,623)(211,631)(212,632)(213,633)(214,634)(215,635)
(216,636)(217,637)(218,645)(219,646)(220,647)(221,648)(222,649)(223,650)
(224,651)(225,638)(226,639)(227,640)(228,641)(229,642)(230,643)(231,644)
(232,652)(233,653)(234,654)(235,655)(236,656)(237,657)(238,658)(239,666)
(240,667)(241,668)(242,669)(243,670)(244,671)(245,672)(246,659)(247,660)
(248,661)(249,662)(250,663)(251,664)(252,665)(253,526)(254,527)(255,528)
(256,529)(257,530)(258,531)(259,532)(260,540)(261,541)(262,542)(263,543)
(264,544)(265,545)(266,546)(267,533)(268,534)(269,535)(270,536)(271,537)
(272,538)(273,539)(274,505)(275,506)(276,507)(277,508)(278,509)(279,510)
(280,511)(281,519)(282,520)(283,521)(284,522)(285,523)(286,524)(287,525)
(288,512)(289,513)(290,514)(291,515)(292,516)(293,517)(294,518)(295,568)
(296,569)(297,570)(298,571)(299,572)(300,573)(301,574)(302,582)(303,583)
(304,584)(305,585)(306,586)(307,587)(308,588)(309,575)(310,576)(311,577)
(312,578)(313,579)(314,580)(315,581)(316,547)(317,548)(318,549)(319,550)
(320,551)(321,552)(322,553)(323,561)(324,562)(325,563)(326,564)(327,565)
(328,566)(329,567)(330,554)(331,555)(332,556)(333,557)(334,558)(335,559)
(336,560);
s2 := Sym(672)!( 1,344)( 2,350)( 3,349)( 4,348)( 5,347)( 6,346)( 7,345)
( 8,337)( 9,343)( 10,342)( 11,341)( 12,340)( 13,339)( 14,338)( 15,351)
( 16,357)( 17,356)( 18,355)( 19,354)( 20,353)( 21,352)( 22,365)( 23,371)
( 24,370)( 25,369)( 26,368)( 27,367)( 28,366)( 29,358)( 30,364)( 31,363)
( 32,362)( 33,361)( 34,360)( 35,359)( 36,372)( 37,378)( 38,377)( 39,376)
( 40,375)( 41,374)( 42,373)( 43,386)( 44,392)( 45,391)( 46,390)( 47,389)
( 48,388)( 49,387)( 50,379)( 51,385)( 52,384)( 53,383)( 54,382)( 55,381)
( 56,380)( 57,393)( 58,399)( 59,398)( 60,397)( 61,396)( 62,395)( 63,394)
( 64,407)( 65,413)( 66,412)( 67,411)( 68,410)( 69,409)( 70,408)( 71,400)
( 72,406)( 73,405)( 74,404)( 75,403)( 76,402)( 77,401)( 78,414)( 79,420)
( 80,419)( 81,418)( 82,417)( 83,416)( 84,415)( 85,449)( 86,455)( 87,454)
( 88,453)( 89,452)( 90,451)( 91,450)( 92,442)( 93,448)( 94,447)( 95,446)
( 96,445)( 97,444)( 98,443)( 99,456)(100,462)(101,461)(102,460)(103,459)
(104,458)(105,457)(106,428)(107,434)(108,433)(109,432)(110,431)(111,430)
(112,429)(113,421)(114,427)(115,426)(116,425)(117,424)(118,423)(119,422)
(120,435)(121,441)(122,440)(123,439)(124,438)(125,437)(126,436)(127,491)
(128,497)(129,496)(130,495)(131,494)(132,493)(133,492)(134,484)(135,490)
(136,489)(137,488)(138,487)(139,486)(140,485)(141,498)(142,504)(143,503)
(144,502)(145,501)(146,500)(147,499)(148,470)(149,476)(150,475)(151,474)
(152,473)(153,472)(154,471)(155,463)(156,469)(157,468)(158,467)(159,466)
(160,465)(161,464)(162,477)(163,483)(164,482)(165,481)(166,480)(167,479)
(168,478)(169,512)(170,518)(171,517)(172,516)(173,515)(174,514)(175,513)
(176,505)(177,511)(178,510)(179,509)(180,508)(181,507)(182,506)(183,519)
(184,525)(185,524)(186,523)(187,522)(188,521)(189,520)(190,533)(191,539)
(192,538)(193,537)(194,536)(195,535)(196,534)(197,526)(198,532)(199,531)
(200,530)(201,529)(202,528)(203,527)(204,540)(205,546)(206,545)(207,544)
(208,543)(209,542)(210,541)(211,554)(212,560)(213,559)(214,558)(215,557)
(216,556)(217,555)(218,547)(219,553)(220,552)(221,551)(222,550)(223,549)
(224,548)(225,561)(226,567)(227,566)(228,565)(229,564)(230,563)(231,562)
(232,575)(233,581)(234,580)(235,579)(236,578)(237,577)(238,576)(239,568)
(240,574)(241,573)(242,572)(243,571)(244,570)(245,569)(246,582)(247,588)
(248,587)(249,586)(250,585)(251,584)(252,583)(253,617)(254,623)(255,622)
(256,621)(257,620)(258,619)(259,618)(260,610)(261,616)(262,615)(263,614)
(264,613)(265,612)(266,611)(267,624)(268,630)(269,629)(270,628)(271,627)
(272,626)(273,625)(274,596)(275,602)(276,601)(277,600)(278,599)(279,598)
(280,597)(281,589)(282,595)(283,594)(284,593)(285,592)(286,591)(287,590)
(288,603)(289,609)(290,608)(291,607)(292,606)(293,605)(294,604)(295,659)
(296,665)(297,664)(298,663)(299,662)(300,661)(301,660)(302,652)(303,658)
(304,657)(305,656)(306,655)(307,654)(308,653)(309,666)(310,672)(311,671)
(312,670)(313,669)(314,668)(315,667)(316,638)(317,644)(318,643)(319,642)
(320,641)(321,640)(322,639)(323,631)(324,637)(325,636)(326,635)(327,634)
(328,633)(329,632)(330,645)(331,651)(332,650)(333,649)(334,648)(335,647)
(336,646);
s3 := Sym(672)!( 1, 2)( 3, 7)( 4, 6)( 8, 9)( 10, 14)( 11, 13)( 15, 16)
( 17, 21)( 18, 20)( 22, 23)( 24, 28)( 25, 27)( 29, 30)( 31, 35)( 32, 34)
( 36, 37)( 38, 42)( 39, 41)( 43, 44)( 45, 49)( 46, 48)( 50, 51)( 52, 56)
( 53, 55)( 57, 58)( 59, 63)( 60, 62)( 64, 65)( 66, 70)( 67, 69)( 71, 72)
( 73, 77)( 74, 76)( 78, 79)( 80, 84)( 81, 83)( 85,107)( 86,106)( 87,112)
( 88,111)( 89,110)( 90,109)( 91,108)( 92,114)( 93,113)( 94,119)( 95,118)
( 96,117)( 97,116)( 98,115)( 99,121)(100,120)(101,126)(102,125)(103,124)
(104,123)(105,122)(127,149)(128,148)(129,154)(130,153)(131,152)(132,151)
(133,150)(134,156)(135,155)(136,161)(137,160)(138,159)(139,158)(140,157)
(141,163)(142,162)(143,168)(144,167)(145,166)(146,165)(147,164)(169,170)
(171,175)(172,174)(176,177)(178,182)(179,181)(183,184)(185,189)(186,188)
(190,191)(192,196)(193,195)(197,198)(199,203)(200,202)(204,205)(206,210)
(207,209)(211,212)(213,217)(214,216)(218,219)(220,224)(221,223)(225,226)
(227,231)(228,230)(232,233)(234,238)(235,237)(239,240)(241,245)(242,244)
(246,247)(248,252)(249,251)(253,275)(254,274)(255,280)(256,279)(257,278)
(258,277)(259,276)(260,282)(261,281)(262,287)(263,286)(264,285)(265,284)
(266,283)(267,289)(268,288)(269,294)(270,293)(271,292)(272,291)(273,290)
(295,317)(296,316)(297,322)(298,321)(299,320)(300,319)(301,318)(302,324)
(303,323)(304,329)(305,328)(306,327)(307,326)(308,325)(309,331)(310,330)
(311,336)(312,335)(313,334)(314,333)(315,332)(337,359)(338,358)(339,364)
(340,363)(341,362)(342,361)(343,360)(344,366)(345,365)(346,371)(347,370)
(348,369)(349,368)(350,367)(351,373)(352,372)(353,378)(354,377)(355,376)
(356,375)(357,374)(379,401)(380,400)(381,406)(382,405)(383,404)(384,403)
(385,402)(386,408)(387,407)(388,413)(389,412)(390,411)(391,410)(392,409)
(393,415)(394,414)(395,420)(396,419)(397,418)(398,417)(399,416)(421,422)
(423,427)(424,426)(428,429)(430,434)(431,433)(435,436)(437,441)(438,440)
(442,443)(444,448)(445,447)(449,450)(451,455)(452,454)(456,457)(458,462)
(459,461)(463,464)(465,469)(466,468)(470,471)(472,476)(473,475)(477,478)
(479,483)(480,482)(484,485)(486,490)(487,489)(491,492)(493,497)(494,496)
(498,499)(500,504)(501,503)(505,527)(506,526)(507,532)(508,531)(509,530)
(510,529)(511,528)(512,534)(513,533)(514,539)(515,538)(516,537)(517,536)
(518,535)(519,541)(520,540)(521,546)(522,545)(523,544)(524,543)(525,542)
(547,569)(548,568)(549,574)(550,573)(551,572)(552,571)(553,570)(554,576)
(555,575)(556,581)(557,580)(558,579)(559,578)(560,577)(561,583)(562,582)
(563,588)(564,587)(565,586)(566,585)(567,584)(589,590)(591,595)(592,594)
(596,597)(598,602)(599,601)(603,604)(605,609)(606,608)(610,611)(612,616)
(613,615)(617,618)(619,623)(620,622)(624,625)(626,630)(627,629)(631,632)
(633,637)(634,636)(638,639)(640,644)(641,643)(645,646)(647,651)(648,650)
(652,653)(654,658)(655,657)(659,660)(661,665)(662,664)(666,667)(668,672)
(669,671);
poly := sub<Sym(672)|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*s3*s2*s1*s2*s3*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*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