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Polytope of Type {84,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {84,6}*1008a
if this polytope has a name.
Group : SmallGroup(1008,599)
Rank : 3
Schlafli Type : {84,6}
Number of vertices, edges, etc : 84, 252, 6
Order of s0s1s2 : 84
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   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 : {42,6}*504a
   3-fold quotients : {28,6}*336a
   6-fold quotients : {14,6}*168
   7-fold quotients : {12,6}*144c
   9-fold quotients : {28,2}*112
   14-fold quotients : {6,6}*72b
   18-fold quotients : {14,2}*56
   21-fold quotients : {4,6}*48a
   28-fold quotients : {6,3}*36
   36-fold quotients : {7,2}*28
   42-fold quotients : {2,6}*24
   63-fold quotients : {4,2}*16
   84-fold quotients : {2,3}*12
   126-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (  1,253)(  2,259)(  3,258)(  4,257)(  5,256)(  6,255)(  7,254)(  8,260)
(  9,266)( 10,265)( 11,264)( 12,263)( 13,262)( 14,261)( 15,267)( 16,273)
( 17,272)( 18,271)( 19,270)( 20,269)( 21,268)( 22,295)( 23,301)( 24,300)
( 25,299)( 26,298)( 27,297)( 28,296)( 29,302)( 30,308)( 31,307)( 32,306)
( 33,305)( 34,304)( 35,303)( 36,309)( 37,315)( 38,314)( 39,313)( 40,312)
( 41,311)( 42,310)( 43,274)( 44,280)( 45,279)( 46,278)( 47,277)( 48,276)
( 49,275)( 50,281)( 51,287)( 52,286)( 53,285)( 54,284)( 55,283)( 56,282)
( 57,288)( 58,294)( 59,293)( 60,292)( 61,291)( 62,290)( 63,289)( 64,316)
( 65,322)( 66,321)( 67,320)( 68,319)( 69,318)( 70,317)( 71,323)( 72,329)
( 73,328)( 74,327)( 75,326)( 76,325)( 77,324)( 78,330)( 79,336)( 80,335)
( 81,334)( 82,333)( 83,332)( 84,331)( 85,358)( 86,364)( 87,363)( 88,362)
( 89,361)( 90,360)( 91,359)( 92,365)( 93,371)( 94,370)( 95,369)( 96,368)
( 97,367)( 98,366)( 99,372)(100,378)(101,377)(102,376)(103,375)(104,374)
(105,373)(106,337)(107,343)(108,342)(109,341)(110,340)(111,339)(112,338)
(113,344)(114,350)(115,349)(116,348)(117,347)(118,346)(119,345)(120,351)
(121,357)(122,356)(123,355)(124,354)(125,353)(126,352)(127,442)(128,448)
(129,447)(130,446)(131,445)(132,444)(133,443)(134,449)(135,455)(136,454)
(137,453)(138,452)(139,451)(140,450)(141,456)(142,462)(143,461)(144,460)
(145,459)(146,458)(147,457)(148,484)(149,490)(150,489)(151,488)(152,487)
(153,486)(154,485)(155,491)(156,497)(157,496)(158,495)(159,494)(160,493)
(161,492)(162,498)(163,504)(164,503)(165,502)(166,501)(167,500)(168,499)
(169,463)(170,469)(171,468)(172,467)(173,466)(174,465)(175,464)(176,470)
(177,476)(178,475)(179,474)(180,473)(181,472)(182,471)(183,477)(184,483)
(185,482)(186,481)(187,480)(188,479)(189,478)(190,379)(191,385)(192,384)
(193,383)(194,382)(195,381)(196,380)(197,386)(198,392)(199,391)(200,390)
(201,389)(202,388)(203,387)(204,393)(205,399)(206,398)(207,397)(208,396)
(209,395)(210,394)(211,421)(212,427)(213,426)(214,425)(215,424)(216,423)
(217,422)(218,428)(219,434)(220,433)(221,432)(222,431)(223,430)(224,429)
(225,435)(226,441)(227,440)(228,439)(229,438)(230,437)(231,436)(232,400)
(233,406)(234,405)(235,404)(236,403)(237,402)(238,401)(239,407)(240,413)
(241,412)(242,411)(243,410)(244,409)(245,408)(246,414)(247,420)(248,419)
(249,418)(250,417)(251,416)(252,415);;
s1 := (  1,401)(  2,400)(  3,406)(  4,405)(  5,404)(  6,403)(  7,402)(  8,415)
(  9,414)( 10,420)( 11,419)( 12,418)( 13,417)( 14,416)( 15,408)( 16,407)
( 17,413)( 18,412)( 19,411)( 20,410)( 21,409)( 22,380)( 23,379)( 24,385)
( 25,384)( 26,383)( 27,382)( 28,381)( 29,394)( 30,393)( 31,399)( 32,398)
( 33,397)( 34,396)( 35,395)( 36,387)( 37,386)( 38,392)( 39,391)( 40,390)
( 41,389)( 42,388)( 43,422)( 44,421)( 45,427)( 46,426)( 47,425)( 48,424)
( 49,423)( 50,436)( 51,435)( 52,441)( 53,440)( 54,439)( 55,438)( 56,437)
( 57,429)( 58,428)( 59,434)( 60,433)( 61,432)( 62,431)( 63,430)( 64,464)
( 65,463)( 66,469)( 67,468)( 68,467)( 69,466)( 70,465)( 71,478)( 72,477)
( 73,483)( 74,482)( 75,481)( 76,480)( 77,479)( 78,471)( 79,470)( 80,476)
( 81,475)( 82,474)( 83,473)( 84,472)( 85,443)( 86,442)( 87,448)( 88,447)
( 89,446)( 90,445)( 91,444)( 92,457)( 93,456)( 94,462)( 95,461)( 96,460)
( 97,459)( 98,458)( 99,450)(100,449)(101,455)(102,454)(103,453)(104,452)
(105,451)(106,485)(107,484)(108,490)(109,489)(110,488)(111,487)(112,486)
(113,499)(114,498)(115,504)(116,503)(117,502)(118,501)(119,500)(120,492)
(121,491)(122,497)(123,496)(124,495)(125,494)(126,493)(127,275)(128,274)
(129,280)(130,279)(131,278)(132,277)(133,276)(134,289)(135,288)(136,294)
(137,293)(138,292)(139,291)(140,290)(141,282)(142,281)(143,287)(144,286)
(145,285)(146,284)(147,283)(148,254)(149,253)(150,259)(151,258)(152,257)
(153,256)(154,255)(155,268)(156,267)(157,273)(158,272)(159,271)(160,270)
(161,269)(162,261)(163,260)(164,266)(165,265)(166,264)(167,263)(168,262)
(169,296)(170,295)(171,301)(172,300)(173,299)(174,298)(175,297)(176,310)
(177,309)(178,315)(179,314)(180,313)(181,312)(182,311)(183,303)(184,302)
(185,308)(186,307)(187,306)(188,305)(189,304)(190,338)(191,337)(192,343)
(193,342)(194,341)(195,340)(196,339)(197,352)(198,351)(199,357)(200,356)
(201,355)(202,354)(203,353)(204,345)(205,344)(206,350)(207,349)(208,348)
(209,347)(210,346)(211,317)(212,316)(213,322)(214,321)(215,320)(216,319)
(217,318)(218,331)(219,330)(220,336)(221,335)(222,334)(223,333)(224,332)
(225,324)(226,323)(227,329)(228,328)(229,327)(230,326)(231,325)(232,359)
(233,358)(234,364)(235,363)(236,362)(237,361)(238,360)(239,373)(240,372)
(241,378)(242,377)(243,376)(244,375)(245,374)(246,366)(247,365)(248,371)
(249,370)(250,369)(251,368)(252,367);;
s2 := (  1,386)(  2,387)(  3,388)(  4,389)(  5,390)(  6,391)(  7,392)(  8,379)
(  9,380)( 10,381)( 11,382)( 12,383)( 13,384)( 14,385)( 15,393)( 16,394)
( 17,395)( 18,396)( 19,397)( 20,398)( 21,399)( 22,428)( 23,429)( 24,430)
( 25,431)( 26,432)( 27,433)( 28,434)( 29,421)( 30,422)( 31,423)( 32,424)
( 33,425)( 34,426)( 35,427)( 36,435)( 37,436)( 38,437)( 39,438)( 40,439)
( 41,440)( 42,441)( 43,407)( 44,408)( 45,409)( 46,410)( 47,411)( 48,412)
( 49,413)( 50,400)( 51,401)( 52,402)( 53,403)( 54,404)( 55,405)( 56,406)
( 57,414)( 58,415)( 59,416)( 60,417)( 61,418)( 62,419)( 63,420)( 64,449)
( 65,450)( 66,451)( 67,452)( 68,453)( 69,454)( 70,455)( 71,442)( 72,443)
( 73,444)( 74,445)( 75,446)( 76,447)( 77,448)( 78,456)( 79,457)( 80,458)
( 81,459)( 82,460)( 83,461)( 84,462)( 85,491)( 86,492)( 87,493)( 88,494)
( 89,495)( 90,496)( 91,497)( 92,484)( 93,485)( 94,486)( 95,487)( 96,488)
( 97,489)( 98,490)( 99,498)(100,499)(101,500)(102,501)(103,502)(104,503)
(105,504)(106,470)(107,471)(108,472)(109,473)(110,474)(111,475)(112,476)
(113,463)(114,464)(115,465)(116,466)(117,467)(118,468)(119,469)(120,477)
(121,478)(122,479)(123,480)(124,481)(125,482)(126,483)(127,323)(128,324)
(129,325)(130,326)(131,327)(132,328)(133,329)(134,316)(135,317)(136,318)
(137,319)(138,320)(139,321)(140,322)(141,330)(142,331)(143,332)(144,333)
(145,334)(146,335)(147,336)(148,365)(149,366)(150,367)(151,368)(152,369)
(153,370)(154,371)(155,358)(156,359)(157,360)(158,361)(159,362)(160,363)
(161,364)(162,372)(163,373)(164,374)(165,375)(166,376)(167,377)(168,378)
(169,344)(170,345)(171,346)(172,347)(173,348)(174,349)(175,350)(176,337)
(177,338)(178,339)(179,340)(180,341)(181,342)(182,343)(183,351)(184,352)
(185,353)(186,354)(187,355)(188,356)(189,357)(190,260)(191,261)(192,262)
(193,263)(194,264)(195,265)(196,266)(197,253)(198,254)(199,255)(200,256)
(201,257)(202,258)(203,259)(204,267)(205,268)(206,269)(207,270)(208,271)
(209,272)(210,273)(211,302)(212,303)(213,304)(214,305)(215,306)(216,307)
(217,308)(218,295)(219,296)(220,297)(221,298)(222,299)(223,300)(224,301)
(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)(232,281)
(233,282)(234,283)(235,284)(236,285)(237,286)(238,287)(239,274)(240,275)
(241,276)(242,277)(243,278)(244,279)(245,280)(246,288)(247,289)(248,290)
(249,291)(250,292)(251,293)(252,294);;
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*s2*s1*s0*s1*s2*s0*s1*s2*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*s2*s0*s1*s0*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*s0*s1 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(504)!(  1,253)(  2,259)(  3,258)(  4,257)(  5,256)(  6,255)(  7,254)
(  8,260)(  9,266)( 10,265)( 11,264)( 12,263)( 13,262)( 14,261)( 15,267)
( 16,273)( 17,272)( 18,271)( 19,270)( 20,269)( 21,268)( 22,295)( 23,301)
( 24,300)( 25,299)( 26,298)( 27,297)( 28,296)( 29,302)( 30,308)( 31,307)
( 32,306)( 33,305)( 34,304)( 35,303)( 36,309)( 37,315)( 38,314)( 39,313)
( 40,312)( 41,311)( 42,310)( 43,274)( 44,280)( 45,279)( 46,278)( 47,277)
( 48,276)( 49,275)( 50,281)( 51,287)( 52,286)( 53,285)( 54,284)( 55,283)
( 56,282)( 57,288)( 58,294)( 59,293)( 60,292)( 61,291)( 62,290)( 63,289)
( 64,316)( 65,322)( 66,321)( 67,320)( 68,319)( 69,318)( 70,317)( 71,323)
( 72,329)( 73,328)( 74,327)( 75,326)( 76,325)( 77,324)( 78,330)( 79,336)
( 80,335)( 81,334)( 82,333)( 83,332)( 84,331)( 85,358)( 86,364)( 87,363)
( 88,362)( 89,361)( 90,360)( 91,359)( 92,365)( 93,371)( 94,370)( 95,369)
( 96,368)( 97,367)( 98,366)( 99,372)(100,378)(101,377)(102,376)(103,375)
(104,374)(105,373)(106,337)(107,343)(108,342)(109,341)(110,340)(111,339)
(112,338)(113,344)(114,350)(115,349)(116,348)(117,347)(118,346)(119,345)
(120,351)(121,357)(122,356)(123,355)(124,354)(125,353)(126,352)(127,442)
(128,448)(129,447)(130,446)(131,445)(132,444)(133,443)(134,449)(135,455)
(136,454)(137,453)(138,452)(139,451)(140,450)(141,456)(142,462)(143,461)
(144,460)(145,459)(146,458)(147,457)(148,484)(149,490)(150,489)(151,488)
(152,487)(153,486)(154,485)(155,491)(156,497)(157,496)(158,495)(159,494)
(160,493)(161,492)(162,498)(163,504)(164,503)(165,502)(166,501)(167,500)
(168,499)(169,463)(170,469)(171,468)(172,467)(173,466)(174,465)(175,464)
(176,470)(177,476)(178,475)(179,474)(180,473)(181,472)(182,471)(183,477)
(184,483)(185,482)(186,481)(187,480)(188,479)(189,478)(190,379)(191,385)
(192,384)(193,383)(194,382)(195,381)(196,380)(197,386)(198,392)(199,391)
(200,390)(201,389)(202,388)(203,387)(204,393)(205,399)(206,398)(207,397)
(208,396)(209,395)(210,394)(211,421)(212,427)(213,426)(214,425)(215,424)
(216,423)(217,422)(218,428)(219,434)(220,433)(221,432)(222,431)(223,430)
(224,429)(225,435)(226,441)(227,440)(228,439)(229,438)(230,437)(231,436)
(232,400)(233,406)(234,405)(235,404)(236,403)(237,402)(238,401)(239,407)
(240,413)(241,412)(242,411)(243,410)(244,409)(245,408)(246,414)(247,420)
(248,419)(249,418)(250,417)(251,416)(252,415);
s1 := Sym(504)!(  1,401)(  2,400)(  3,406)(  4,405)(  5,404)(  6,403)(  7,402)
(  8,415)(  9,414)( 10,420)( 11,419)( 12,418)( 13,417)( 14,416)( 15,408)
( 16,407)( 17,413)( 18,412)( 19,411)( 20,410)( 21,409)( 22,380)( 23,379)
( 24,385)( 25,384)( 26,383)( 27,382)( 28,381)( 29,394)( 30,393)( 31,399)
( 32,398)( 33,397)( 34,396)( 35,395)( 36,387)( 37,386)( 38,392)( 39,391)
( 40,390)( 41,389)( 42,388)( 43,422)( 44,421)( 45,427)( 46,426)( 47,425)
( 48,424)( 49,423)( 50,436)( 51,435)( 52,441)( 53,440)( 54,439)( 55,438)
( 56,437)( 57,429)( 58,428)( 59,434)( 60,433)( 61,432)( 62,431)( 63,430)
( 64,464)( 65,463)( 66,469)( 67,468)( 68,467)( 69,466)( 70,465)( 71,478)
( 72,477)( 73,483)( 74,482)( 75,481)( 76,480)( 77,479)( 78,471)( 79,470)
( 80,476)( 81,475)( 82,474)( 83,473)( 84,472)( 85,443)( 86,442)( 87,448)
( 88,447)( 89,446)( 90,445)( 91,444)( 92,457)( 93,456)( 94,462)( 95,461)
( 96,460)( 97,459)( 98,458)( 99,450)(100,449)(101,455)(102,454)(103,453)
(104,452)(105,451)(106,485)(107,484)(108,490)(109,489)(110,488)(111,487)
(112,486)(113,499)(114,498)(115,504)(116,503)(117,502)(118,501)(119,500)
(120,492)(121,491)(122,497)(123,496)(124,495)(125,494)(126,493)(127,275)
(128,274)(129,280)(130,279)(131,278)(132,277)(133,276)(134,289)(135,288)
(136,294)(137,293)(138,292)(139,291)(140,290)(141,282)(142,281)(143,287)
(144,286)(145,285)(146,284)(147,283)(148,254)(149,253)(150,259)(151,258)
(152,257)(153,256)(154,255)(155,268)(156,267)(157,273)(158,272)(159,271)
(160,270)(161,269)(162,261)(163,260)(164,266)(165,265)(166,264)(167,263)
(168,262)(169,296)(170,295)(171,301)(172,300)(173,299)(174,298)(175,297)
(176,310)(177,309)(178,315)(179,314)(180,313)(181,312)(182,311)(183,303)
(184,302)(185,308)(186,307)(187,306)(188,305)(189,304)(190,338)(191,337)
(192,343)(193,342)(194,341)(195,340)(196,339)(197,352)(198,351)(199,357)
(200,356)(201,355)(202,354)(203,353)(204,345)(205,344)(206,350)(207,349)
(208,348)(209,347)(210,346)(211,317)(212,316)(213,322)(214,321)(215,320)
(216,319)(217,318)(218,331)(219,330)(220,336)(221,335)(222,334)(223,333)
(224,332)(225,324)(226,323)(227,329)(228,328)(229,327)(230,326)(231,325)
(232,359)(233,358)(234,364)(235,363)(236,362)(237,361)(238,360)(239,373)
(240,372)(241,378)(242,377)(243,376)(244,375)(245,374)(246,366)(247,365)
(248,371)(249,370)(250,369)(251,368)(252,367);
s2 := Sym(504)!(  1,386)(  2,387)(  3,388)(  4,389)(  5,390)(  6,391)(  7,392)
(  8,379)(  9,380)( 10,381)( 11,382)( 12,383)( 13,384)( 14,385)( 15,393)
( 16,394)( 17,395)( 18,396)( 19,397)( 20,398)( 21,399)( 22,428)( 23,429)
( 24,430)( 25,431)( 26,432)( 27,433)( 28,434)( 29,421)( 30,422)( 31,423)
( 32,424)( 33,425)( 34,426)( 35,427)( 36,435)( 37,436)( 38,437)( 39,438)
( 40,439)( 41,440)( 42,441)( 43,407)( 44,408)( 45,409)( 46,410)( 47,411)
( 48,412)( 49,413)( 50,400)( 51,401)( 52,402)( 53,403)( 54,404)( 55,405)
( 56,406)( 57,414)( 58,415)( 59,416)( 60,417)( 61,418)( 62,419)( 63,420)
( 64,449)( 65,450)( 66,451)( 67,452)( 68,453)( 69,454)( 70,455)( 71,442)
( 72,443)( 73,444)( 74,445)( 75,446)( 76,447)( 77,448)( 78,456)( 79,457)
( 80,458)( 81,459)( 82,460)( 83,461)( 84,462)( 85,491)( 86,492)( 87,493)
( 88,494)( 89,495)( 90,496)( 91,497)( 92,484)( 93,485)( 94,486)( 95,487)
( 96,488)( 97,489)( 98,490)( 99,498)(100,499)(101,500)(102,501)(103,502)
(104,503)(105,504)(106,470)(107,471)(108,472)(109,473)(110,474)(111,475)
(112,476)(113,463)(114,464)(115,465)(116,466)(117,467)(118,468)(119,469)
(120,477)(121,478)(122,479)(123,480)(124,481)(125,482)(126,483)(127,323)
(128,324)(129,325)(130,326)(131,327)(132,328)(133,329)(134,316)(135,317)
(136,318)(137,319)(138,320)(139,321)(140,322)(141,330)(142,331)(143,332)
(144,333)(145,334)(146,335)(147,336)(148,365)(149,366)(150,367)(151,368)
(152,369)(153,370)(154,371)(155,358)(156,359)(157,360)(158,361)(159,362)
(160,363)(161,364)(162,372)(163,373)(164,374)(165,375)(166,376)(167,377)
(168,378)(169,344)(170,345)(171,346)(172,347)(173,348)(174,349)(175,350)
(176,337)(177,338)(178,339)(179,340)(180,341)(181,342)(182,343)(183,351)
(184,352)(185,353)(186,354)(187,355)(188,356)(189,357)(190,260)(191,261)
(192,262)(193,263)(194,264)(195,265)(196,266)(197,253)(198,254)(199,255)
(200,256)(201,257)(202,258)(203,259)(204,267)(205,268)(206,269)(207,270)
(208,271)(209,272)(210,273)(211,302)(212,303)(213,304)(214,305)(215,306)
(216,307)(217,308)(218,295)(219,296)(220,297)(221,298)(222,299)(223,300)
(224,301)(225,309)(226,310)(227,311)(228,312)(229,313)(230,314)(231,315)
(232,281)(233,282)(234,283)(235,284)(236,285)(237,286)(238,287)(239,274)
(240,275)(241,276)(242,277)(243,278)(244,279)(245,280)(246,288)(247,289)
(248,290)(249,291)(250,292)(251,293)(252,294);
poly := sub<Sym(504)|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*s2*s1*s0*s1*s2*s0*s1*s2*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*s2*s0*s1*s0*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*s0*s1 >; 
 
References : None.
to this polytope