Polytope of Type {12,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {12,6}*1296i
if this polytope has a name.
Group : SmallGroup(1296,2077)
Rank : 3
Schlafli Type : {12,6}
Number of vertices, edges, etc : 108, 324, 54
Order of s₀s₁s₂ : 12
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   None in this Atlas
Vertex Figure Of :
   None in this Atlas
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,6}*648f
   3-fold quotients : {12,6}*432c, {12,6}*432g
   6-fold quotients : {6,6}*216a, {6,6}*216d
   9-fold quotients : {12,6}*144a, {12,6}*144b, {12,6}*144c
   12-fold quotients : {6,3}*108
   18-fold quotients : {6,6}*72a, {6,6}*72b, {6,6}*72c
   27-fold quotients : {12,2}*48, {4,6}*48a
   36-fold quotients : {3,6}*36, {6,3}*36
   54-fold quotients : {2,6}*24, {6,2}*24
   81-fold quotients : {4,2}*16
   108-fold quotients : {2,3}*12, {3,2}*12
   162-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :

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

Finitely Presented Group Representation (GAP) :

F := FreeGroup("s0","s1","s2");;
s0 := F.1;;  s1 := F.2;;  s2 := F.3;;  
rels := [ s0*s0, s1*s1, s2*s2, s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 ];;
poly := F / rels;;

Permutation Representation (Magma) :

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

Finitely Presented Group Representation (Magma) :

poly<s0,s1,s2> := Group< s0,s1,s2 | s0*s0, s1*s1, s2*s2, 
s0*s2*s0*s2, s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2, 
s2*s0*s1*s0*s1*s2*s1*s0*s1*s2*s0*s1*s0*s1*s2*s1*s0*s1, 
s2*s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1*s0*s1, 
s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1 >;

References : None.
to this polytope