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# Polytope of Type {6,84}

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

```
Permutation Representation (Magma) :
```s0 := Sym(504)!(  1,379)(  2,380)(  3,381)(  4,382)(  5,383)(  6,384)(  7,385)
(  8,393)(  9,394)( 10,395)( 11,396)( 12,397)( 13,398)( 14,399)( 15,386)
( 16,387)( 17,388)( 18,389)( 19,390)( 20,391)( 21,392)( 22,421)( 23,422)
( 24,423)( 25,424)( 26,425)( 27,426)( 28,427)( 29,435)( 30,436)( 31,437)
( 32,438)( 33,439)( 34,440)( 35,441)( 36,428)( 37,429)( 38,430)( 39,431)
( 40,432)( 41,433)( 42,434)( 43,400)( 44,401)( 45,402)( 46,403)( 47,404)
( 48,405)( 49,406)( 50,414)( 51,415)( 52,416)( 53,417)( 54,418)( 55,419)
( 56,420)( 57,407)( 58,408)( 59,409)( 60,410)( 61,411)( 62,412)( 63,413)
( 64,442)( 65,443)( 66,444)( 67,445)( 68,446)( 69,447)( 70,448)( 71,456)
( 72,457)( 73,458)( 74,459)( 75,460)( 76,461)( 77,462)( 78,449)( 79,450)
( 80,451)( 81,452)( 82,453)( 83,454)( 84,455)( 85,484)( 86,485)( 87,486)
( 88,487)( 89,488)( 90,489)( 91,490)( 92,498)( 93,499)( 94,500)( 95,501)
( 96,502)( 97,503)( 98,504)( 99,491)(100,492)(101,493)(102,494)(103,495)
(104,496)(105,497)(106,463)(107,464)(108,465)(109,466)(110,467)(111,468)
(112,469)(113,477)(114,478)(115,479)(116,480)(117,481)(118,482)(119,483)
(120,470)(121,471)(122,472)(123,473)(124,474)(125,475)(126,476)(127,316)
(128,317)(129,318)(130,319)(131,320)(132,321)(133,322)(134,330)(135,331)
(136,332)(137,333)(138,334)(139,335)(140,336)(141,323)(142,324)(143,325)
(144,326)(145,327)(146,328)(147,329)(148,358)(149,359)(150,360)(151,361)
(152,362)(153,363)(154,364)(155,372)(156,373)(157,374)(158,375)(159,376)
(160,377)(161,378)(162,365)(163,366)(164,367)(165,368)(166,369)(167,370)
(168,371)(169,337)(170,338)(171,339)(172,340)(173,341)(174,342)(175,343)
(176,351)(177,352)(178,353)(179,354)(180,355)(181,356)(182,357)(183,344)
(184,345)(185,346)(186,347)(187,348)(188,349)(189,350)(190,253)(191,254)
(192,255)(193,256)(194,257)(195,258)(196,259)(197,267)(198,268)(199,269)
(200,270)(201,271)(202,272)(203,273)(204,260)(205,261)(206,262)(207,263)
(208,264)(209,265)(210,266)(211,295)(212,296)(213,297)(214,298)(215,299)
(216,300)(217,301)(218,309)(219,310)(220,311)(221,312)(222,313)(223,314)
(224,315)(225,302)(226,303)(227,304)(228,305)(229,306)(230,307)(231,308)
(232,274)(233,275)(234,276)(235,277)(236,278)(237,279)(238,280)(239,288)
(240,289)(241,290)(242,291)(243,292)(244,293)(245,294)(246,281)(247,282)
(248,283)(249,284)(250,285)(251,286)(252,287);
s1 := Sym(504)!(  1,407)(  2,413)(  3,412)(  4,411)(  5,410)(  6,409)(  7,408)
(  8,400)(  9,406)( 10,405)( 11,404)( 12,403)( 13,402)( 14,401)( 15,414)
( 16,420)( 17,419)( 18,418)( 19,417)( 20,416)( 21,415)( 22,386)( 23,392)
( 24,391)( 25,390)( 26,389)( 27,388)( 28,387)( 29,379)( 30,385)( 31,384)
( 32,383)( 33,382)( 34,381)( 35,380)( 36,393)( 37,399)( 38,398)( 39,397)
( 40,396)( 41,395)( 42,394)( 43,428)( 44,434)( 45,433)( 46,432)( 47,431)
( 48,430)( 49,429)( 50,421)( 51,427)( 52,426)( 53,425)( 54,424)( 55,423)
( 56,422)( 57,435)( 58,441)( 59,440)( 60,439)( 61,438)( 62,437)( 63,436)
( 64,470)( 65,476)( 66,475)( 67,474)( 68,473)( 69,472)( 70,471)( 71,463)
( 72,469)( 73,468)( 74,467)( 75,466)( 76,465)( 77,464)( 78,477)( 79,483)
( 80,482)( 81,481)( 82,480)( 83,479)( 84,478)( 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)(106,491)(107,497)(108,496)(109,495)(110,494)(111,493)
(112,492)(113,484)(114,490)(115,489)(116,488)(117,487)(118,486)(119,485)
(120,498)(121,504)(122,503)(123,502)(124,501)(125,500)(126,499)(127,281)
(128,287)(129,286)(130,285)(131,284)(132,283)(133,282)(134,274)(135,280)
(136,279)(137,278)(138,277)(139,276)(140,275)(141,288)(142,294)(143,293)
(144,292)(145,291)(146,290)(147,289)(148,260)(149,266)(150,265)(151,264)
(152,263)(153,262)(154,261)(155,253)(156,259)(157,258)(158,257)(159,256)
(160,255)(161,254)(162,267)(163,273)(164,272)(165,271)(166,270)(167,269)
(168,268)(169,302)(170,308)(171,307)(172,306)(173,305)(174,304)(175,303)
(176,295)(177,301)(178,300)(179,299)(180,298)(181,297)(182,296)(183,309)
(184,315)(185,314)(186,313)(187,312)(188,311)(189,310)(190,344)(191,350)
(192,349)(193,348)(194,347)(195,346)(196,345)(197,337)(198,343)(199,342)
(200,341)(201,340)(202,339)(203,338)(204,351)(205,357)(206,356)(207,355)
(208,354)(209,353)(210,352)(211,323)(212,329)(213,328)(214,327)(215,326)
(216,325)(217,324)(218,316)(219,322)(220,321)(221,320)(222,319)(223,318)
(224,317)(225,330)(226,336)(227,335)(228,334)(229,333)(230,332)(231,331)
(232,365)(233,371)(234,370)(235,369)(236,368)(237,367)(238,366)(239,358)
(240,364)(241,363)(242,362)(243,361)(244,360)(245,359)(246,372)(247,378)
(248,377)(249,376)(250,375)(251,374)(252,373);
s2 := Sym(504)!(  1,254)(  2,253)(  3,259)(  4,258)(  5,257)(  6,256)(  7,255)
(  8,261)(  9,260)( 10,266)( 11,265)( 12,264)( 13,263)( 14,262)( 15,268)
( 16,267)( 17,273)( 18,272)( 19,271)( 20,270)( 21,269)( 22,296)( 23,295)
( 24,301)( 25,300)( 26,299)( 27,298)( 28,297)( 29,303)( 30,302)( 31,308)
( 32,307)( 33,306)( 34,305)( 35,304)( 36,310)( 37,309)( 38,315)( 39,314)
( 40,313)( 41,312)( 42,311)( 43,275)( 44,274)( 45,280)( 46,279)( 47,278)
( 48,277)( 49,276)( 50,282)( 51,281)( 52,287)( 53,286)( 54,285)( 55,284)
( 56,283)( 57,289)( 58,288)( 59,294)( 60,293)( 61,292)( 62,291)( 63,290)
( 64,317)( 65,316)( 66,322)( 67,321)( 68,320)( 69,319)( 70,318)( 71,324)
( 72,323)( 73,329)( 74,328)( 75,327)( 76,326)( 77,325)( 78,331)( 79,330)
( 80,336)( 81,335)( 82,334)( 83,333)( 84,332)( 85,359)( 86,358)( 87,364)
( 88,363)( 89,362)( 90,361)( 91,360)( 92,366)( 93,365)( 94,371)( 95,370)
( 96,369)( 97,368)( 98,367)( 99,373)(100,372)(101,378)(102,377)(103,376)
(104,375)(105,374)(106,338)(107,337)(108,343)(109,342)(110,341)(111,340)
(112,339)(113,345)(114,344)(115,350)(116,349)(117,348)(118,347)(119,346)
(120,352)(121,351)(122,357)(123,356)(124,355)(125,354)(126,353)(127,443)
(128,442)(129,448)(130,447)(131,446)(132,445)(133,444)(134,450)(135,449)
(136,455)(137,454)(138,453)(139,452)(140,451)(141,457)(142,456)(143,462)
(144,461)(145,460)(146,459)(147,458)(148,485)(149,484)(150,490)(151,489)
(152,488)(153,487)(154,486)(155,492)(156,491)(157,497)(158,496)(159,495)
(160,494)(161,493)(162,499)(163,498)(164,504)(165,503)(166,502)(167,501)
(168,500)(169,464)(170,463)(171,469)(172,468)(173,467)(174,466)(175,465)
(176,471)(177,470)(178,476)(179,475)(180,474)(181,473)(182,472)(183,478)
(184,477)(185,483)(186,482)(187,481)(188,480)(189,479)(190,380)(191,379)
(192,385)(193,384)(194,383)(195,382)(196,381)(197,387)(198,386)(199,392)
(200,391)(201,390)(202,389)(203,388)(204,394)(205,393)(206,399)(207,398)
(208,397)(209,396)(210,395)(211,422)(212,421)(213,427)(214,426)(215,425)
(216,424)(217,423)(218,429)(219,428)(220,434)(221,433)(222,432)(223,431)
(224,430)(225,436)(226,435)(227,441)(228,440)(229,439)(230,438)(231,437)
(232,401)(233,400)(234,406)(235,405)(236,404)(237,403)(238,402)(239,408)
(240,407)(241,413)(242,412)(243,411)(244,410)(245,409)(246,415)(247,414)
(248,420)(249,419)(250,418)(251,417)(252,416);
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, s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1,
s2*s0*s1*s2*s1*s0*s1*s2*s0*s1*s2*s1*s0*s1,
s2*s0*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1 >;

```
References : None.
to this polytope