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Polytope of Type {2,24,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {2,24,6}*1728f
if this polytope has a name.
Group : SmallGroup(1728,36607)
Rank : 4
Schlafli Type : {2,24,6}
Number of vertices, edges, etc : 2, 72, 216, 18
Order of s0s1s2s3 : 24
Order of s0s1s2s3s2s1 : 2
Special Properties :
   Degenerate
   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 : {2,12,6}*864g
   3-fold quotients : {2,24,6}*576a, {2,24,6}*576b, {2,24,6}*576c
   4-fold quotients : {2,6,6}*432d
   6-fold quotients : {2,12,6}*288a, {2,12,6}*288b, {2,12,6}*288c
   9-fold quotients : {2,24,2}*192, {2,8,6}*192
   12-fold quotients : {2,6,6}*144a, {2,6,6}*144b, {2,6,6}*144c
   18-fold quotients : {2,12,2}*96, {2,4,6}*96a
   24-fold quotients : {2,3,6}*72, {2,6,3}*72
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,2,6}*48, {2,6,2}*48
   54-fold quotients : {2,4,2}*32
   72-fold quotients : {2,2,3}*24, {2,3,2}*24
   108-fold quotients : {2,2,2}*16
Covers (Minimal Covers in Boldface) :
   None in this atlas.
Permutation Representation (GAP) :
s0 := (1,2);;
s1 := (  3,219)(  4,221)(  5,220)(  6,225)(  7,227)(  8,226)(  9,222)( 10,224)
( 11,223)( 12,228)( 13,230)( 14,229)( 15,234)( 16,236)( 17,235)( 18,231)
( 19,233)( 20,232)( 21,237)( 22,239)( 23,238)( 24,243)( 25,245)( 26,244)
( 27,240)( 28,242)( 29,241)( 30,246)( 31,248)( 32,247)( 33,252)( 34,254)
( 35,253)( 36,249)( 37,251)( 38,250)( 39,255)( 40,257)( 41,256)( 42,261)
( 43,263)( 44,262)( 45,258)( 46,260)( 47,259)( 48,264)( 49,266)( 50,265)
( 51,270)( 52,272)( 53,271)( 54,267)( 55,269)( 56,268)( 57,300)( 58,302)
( 59,301)( 60,306)( 61,308)( 62,307)( 63,303)( 64,305)( 65,304)( 66,309)
( 67,311)( 68,310)( 69,315)( 70,317)( 71,316)( 72,312)( 73,314)( 74,313)
( 75,318)( 76,320)( 77,319)( 78,324)( 79,326)( 80,325)( 81,321)( 82,323)
( 83,322)( 84,273)( 85,275)( 86,274)( 87,279)( 88,281)( 89,280)( 90,276)
( 91,278)( 92,277)( 93,282)( 94,284)( 95,283)( 96,288)( 97,290)( 98,289)
( 99,285)(100,287)(101,286)(102,291)(103,293)(104,292)(105,297)(106,299)
(107,298)(108,294)(109,296)(110,295)(111,381)(112,383)(113,382)(114,387)
(115,389)(116,388)(117,384)(118,386)(119,385)(120,390)(121,392)(122,391)
(123,396)(124,398)(125,397)(126,393)(127,395)(128,394)(129,399)(130,401)
(131,400)(132,405)(133,407)(134,406)(135,402)(136,404)(137,403)(138,408)
(139,410)(140,409)(141,414)(142,416)(143,415)(144,411)(145,413)(146,412)
(147,417)(148,419)(149,418)(150,423)(151,425)(152,424)(153,420)(154,422)
(155,421)(156,426)(157,428)(158,427)(159,432)(160,434)(161,433)(162,429)
(163,431)(164,430)(165,327)(166,329)(167,328)(168,333)(169,335)(170,334)
(171,330)(172,332)(173,331)(174,336)(175,338)(176,337)(177,342)(178,344)
(179,343)(180,339)(181,341)(182,340)(183,345)(184,347)(185,346)(186,351)
(187,353)(188,352)(189,348)(190,350)(191,349)(192,354)(193,356)(194,355)
(195,360)(196,362)(197,361)(198,357)(199,359)(200,358)(201,363)(202,365)
(203,364)(204,369)(205,371)(206,370)(207,366)(208,368)(209,367)(210,372)
(211,374)(212,373)(213,378)(214,380)(215,379)(216,375)(217,377)(218,376);;
s2 := (  3,331)(  4,330)(  5,332)(  6,328)(  7,327)(  8,329)(  9,334)( 10,333)
( 11,335)( 12,349)( 13,348)( 14,350)( 15,346)( 16,345)( 17,347)( 18,352)
( 19,351)( 20,353)( 21,340)( 22,339)( 23,341)( 24,337)( 25,336)( 26,338)
( 27,343)( 28,342)( 29,344)( 30,358)( 31,357)( 32,359)( 33,355)( 34,354)
( 35,356)( 36,361)( 37,360)( 38,362)( 39,376)( 40,375)( 41,377)( 42,373)
( 43,372)( 44,374)( 45,379)( 46,378)( 47,380)( 48,367)( 49,366)( 50,368)
( 51,364)( 52,363)( 53,365)( 54,370)( 55,369)( 56,371)( 57,412)( 58,411)
( 59,413)( 60,409)( 61,408)( 62,410)( 63,415)( 64,414)( 65,416)( 66,430)
( 67,429)( 68,431)( 69,427)( 70,426)( 71,428)( 72,433)( 73,432)( 74,434)
( 75,421)( 76,420)( 77,422)( 78,418)( 79,417)( 80,419)( 81,424)( 82,423)
( 83,425)( 84,385)( 85,384)( 86,386)( 87,382)( 88,381)( 89,383)( 90,388)
( 91,387)( 92,389)( 93,403)( 94,402)( 95,404)( 96,400)( 97,399)( 98,401)
( 99,406)(100,405)(101,407)(102,394)(103,393)(104,395)(105,391)(106,390)
(107,392)(108,397)(109,396)(110,398)(111,223)(112,222)(113,224)(114,220)
(115,219)(116,221)(117,226)(118,225)(119,227)(120,241)(121,240)(122,242)
(123,238)(124,237)(125,239)(126,244)(127,243)(128,245)(129,232)(130,231)
(131,233)(132,229)(133,228)(134,230)(135,235)(136,234)(137,236)(138,250)
(139,249)(140,251)(141,247)(142,246)(143,248)(144,253)(145,252)(146,254)
(147,268)(148,267)(149,269)(150,265)(151,264)(152,266)(153,271)(154,270)
(155,272)(156,259)(157,258)(158,260)(159,256)(160,255)(161,257)(162,262)
(163,261)(164,263)(165,304)(166,303)(167,305)(168,301)(169,300)(170,302)
(171,307)(172,306)(173,308)(174,322)(175,321)(176,323)(177,319)(178,318)
(179,320)(180,325)(181,324)(182,326)(183,313)(184,312)(185,314)(186,310)
(187,309)(188,311)(189,316)(190,315)(191,317)(192,277)(193,276)(194,278)
(195,274)(196,273)(197,275)(198,280)(199,279)(200,281)(201,295)(202,294)
(203,296)(204,292)(205,291)(206,293)(207,298)(208,297)(209,299)(210,286)
(211,285)(212,287)(213,283)(214,282)(215,284)(216,289)(217,288)(218,290);;
s3 := (  3,390)(  4,391)(  5,392)(  6,396)(  7,397)(  8,398)(  9,393)( 10,394)
( 11,395)( 12,381)( 13,382)( 14,383)( 15,387)( 16,388)( 17,389)( 18,384)
( 19,385)( 20,386)( 21,399)( 22,400)( 23,401)( 24,405)( 25,406)( 26,407)
( 27,402)( 28,403)( 29,404)( 30,417)( 31,418)( 32,419)( 33,423)( 34,424)
( 35,425)( 36,420)( 37,421)( 38,422)( 39,408)( 40,409)( 41,410)( 42,414)
( 43,415)( 44,416)( 45,411)( 46,412)( 47,413)( 48,426)( 49,427)( 50,428)
( 51,432)( 52,433)( 53,434)( 54,429)( 55,430)( 56,431)( 57,363)( 58,364)
( 59,365)( 60,369)( 61,370)( 62,371)( 63,366)( 64,367)( 65,368)( 66,354)
( 67,355)( 68,356)( 69,360)( 70,361)( 71,362)( 72,357)( 73,358)( 74,359)
( 75,372)( 76,373)( 77,374)( 78,378)( 79,379)( 80,380)( 81,375)( 82,376)
( 83,377)( 84,336)( 85,337)( 86,338)( 87,342)( 88,343)( 89,344)( 90,339)
( 91,340)( 92,341)( 93,327)( 94,328)( 95,329)( 96,333)( 97,334)( 98,335)
( 99,330)(100,331)(101,332)(102,345)(103,346)(104,347)(105,351)(106,352)
(107,353)(108,348)(109,349)(110,350)(111,228)(112,229)(113,230)(114,234)
(115,235)(116,236)(117,231)(118,232)(119,233)(120,219)(121,220)(122,221)
(123,225)(124,226)(125,227)(126,222)(127,223)(128,224)(129,237)(130,238)
(131,239)(132,243)(133,244)(134,245)(135,240)(136,241)(137,242)(138,255)
(139,256)(140,257)(141,261)(142,262)(143,263)(144,258)(145,259)(146,260)
(147,246)(148,247)(149,248)(150,252)(151,253)(152,254)(153,249)(154,250)
(155,251)(156,264)(157,265)(158,266)(159,270)(160,271)(161,272)(162,267)
(163,268)(164,269)(165,282)(166,283)(167,284)(168,288)(169,289)(170,290)
(171,285)(172,286)(173,287)(174,273)(175,274)(176,275)(177,279)(178,280)
(179,281)(180,276)(181,277)(182,278)(183,291)(184,292)(185,293)(186,297)
(187,298)(188,299)(189,294)(190,295)(191,296)(192,309)(193,310)(194,311)
(195,315)(196,316)(197,317)(198,312)(199,313)(200,314)(201,300)(202,301)
(203,302)(204,306)(205,307)(206,308)(207,303)(208,304)(209,305)(210,318)
(211,319)(212,320)(213,324)(214,325)(215,326)(216,321)(217,322)(218,323);;
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*s1*s0*s1, 
s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3, 
s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*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 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(434)!(1,2);
s1 := Sym(434)!(  3,219)(  4,221)(  5,220)(  6,225)(  7,227)(  8,226)(  9,222)
( 10,224)( 11,223)( 12,228)( 13,230)( 14,229)( 15,234)( 16,236)( 17,235)
( 18,231)( 19,233)( 20,232)( 21,237)( 22,239)( 23,238)( 24,243)( 25,245)
( 26,244)( 27,240)( 28,242)( 29,241)( 30,246)( 31,248)( 32,247)( 33,252)
( 34,254)( 35,253)( 36,249)( 37,251)( 38,250)( 39,255)( 40,257)( 41,256)
( 42,261)( 43,263)( 44,262)( 45,258)( 46,260)( 47,259)( 48,264)( 49,266)
( 50,265)( 51,270)( 52,272)( 53,271)( 54,267)( 55,269)( 56,268)( 57,300)
( 58,302)( 59,301)( 60,306)( 61,308)( 62,307)( 63,303)( 64,305)( 65,304)
( 66,309)( 67,311)( 68,310)( 69,315)( 70,317)( 71,316)( 72,312)( 73,314)
( 74,313)( 75,318)( 76,320)( 77,319)( 78,324)( 79,326)( 80,325)( 81,321)
( 82,323)( 83,322)( 84,273)( 85,275)( 86,274)( 87,279)( 88,281)( 89,280)
( 90,276)( 91,278)( 92,277)( 93,282)( 94,284)( 95,283)( 96,288)( 97,290)
( 98,289)( 99,285)(100,287)(101,286)(102,291)(103,293)(104,292)(105,297)
(106,299)(107,298)(108,294)(109,296)(110,295)(111,381)(112,383)(113,382)
(114,387)(115,389)(116,388)(117,384)(118,386)(119,385)(120,390)(121,392)
(122,391)(123,396)(124,398)(125,397)(126,393)(127,395)(128,394)(129,399)
(130,401)(131,400)(132,405)(133,407)(134,406)(135,402)(136,404)(137,403)
(138,408)(139,410)(140,409)(141,414)(142,416)(143,415)(144,411)(145,413)
(146,412)(147,417)(148,419)(149,418)(150,423)(151,425)(152,424)(153,420)
(154,422)(155,421)(156,426)(157,428)(158,427)(159,432)(160,434)(161,433)
(162,429)(163,431)(164,430)(165,327)(166,329)(167,328)(168,333)(169,335)
(170,334)(171,330)(172,332)(173,331)(174,336)(175,338)(176,337)(177,342)
(178,344)(179,343)(180,339)(181,341)(182,340)(183,345)(184,347)(185,346)
(186,351)(187,353)(188,352)(189,348)(190,350)(191,349)(192,354)(193,356)
(194,355)(195,360)(196,362)(197,361)(198,357)(199,359)(200,358)(201,363)
(202,365)(203,364)(204,369)(205,371)(206,370)(207,366)(208,368)(209,367)
(210,372)(211,374)(212,373)(213,378)(214,380)(215,379)(216,375)(217,377)
(218,376);
s2 := Sym(434)!(  3,331)(  4,330)(  5,332)(  6,328)(  7,327)(  8,329)(  9,334)
( 10,333)( 11,335)( 12,349)( 13,348)( 14,350)( 15,346)( 16,345)( 17,347)
( 18,352)( 19,351)( 20,353)( 21,340)( 22,339)( 23,341)( 24,337)( 25,336)
( 26,338)( 27,343)( 28,342)( 29,344)( 30,358)( 31,357)( 32,359)( 33,355)
( 34,354)( 35,356)( 36,361)( 37,360)( 38,362)( 39,376)( 40,375)( 41,377)
( 42,373)( 43,372)( 44,374)( 45,379)( 46,378)( 47,380)( 48,367)( 49,366)
( 50,368)( 51,364)( 52,363)( 53,365)( 54,370)( 55,369)( 56,371)( 57,412)
( 58,411)( 59,413)( 60,409)( 61,408)( 62,410)( 63,415)( 64,414)( 65,416)
( 66,430)( 67,429)( 68,431)( 69,427)( 70,426)( 71,428)( 72,433)( 73,432)
( 74,434)( 75,421)( 76,420)( 77,422)( 78,418)( 79,417)( 80,419)( 81,424)
( 82,423)( 83,425)( 84,385)( 85,384)( 86,386)( 87,382)( 88,381)( 89,383)
( 90,388)( 91,387)( 92,389)( 93,403)( 94,402)( 95,404)( 96,400)( 97,399)
( 98,401)( 99,406)(100,405)(101,407)(102,394)(103,393)(104,395)(105,391)
(106,390)(107,392)(108,397)(109,396)(110,398)(111,223)(112,222)(113,224)
(114,220)(115,219)(116,221)(117,226)(118,225)(119,227)(120,241)(121,240)
(122,242)(123,238)(124,237)(125,239)(126,244)(127,243)(128,245)(129,232)
(130,231)(131,233)(132,229)(133,228)(134,230)(135,235)(136,234)(137,236)
(138,250)(139,249)(140,251)(141,247)(142,246)(143,248)(144,253)(145,252)
(146,254)(147,268)(148,267)(149,269)(150,265)(151,264)(152,266)(153,271)
(154,270)(155,272)(156,259)(157,258)(158,260)(159,256)(160,255)(161,257)
(162,262)(163,261)(164,263)(165,304)(166,303)(167,305)(168,301)(169,300)
(170,302)(171,307)(172,306)(173,308)(174,322)(175,321)(176,323)(177,319)
(178,318)(179,320)(180,325)(181,324)(182,326)(183,313)(184,312)(185,314)
(186,310)(187,309)(188,311)(189,316)(190,315)(191,317)(192,277)(193,276)
(194,278)(195,274)(196,273)(197,275)(198,280)(199,279)(200,281)(201,295)
(202,294)(203,296)(204,292)(205,291)(206,293)(207,298)(208,297)(209,299)
(210,286)(211,285)(212,287)(213,283)(214,282)(215,284)(216,289)(217,288)
(218,290);
s3 := Sym(434)!(  3,390)(  4,391)(  5,392)(  6,396)(  7,397)(  8,398)(  9,393)
( 10,394)( 11,395)( 12,381)( 13,382)( 14,383)( 15,387)( 16,388)( 17,389)
( 18,384)( 19,385)( 20,386)( 21,399)( 22,400)( 23,401)( 24,405)( 25,406)
( 26,407)( 27,402)( 28,403)( 29,404)( 30,417)( 31,418)( 32,419)( 33,423)
( 34,424)( 35,425)( 36,420)( 37,421)( 38,422)( 39,408)( 40,409)( 41,410)
( 42,414)( 43,415)( 44,416)( 45,411)( 46,412)( 47,413)( 48,426)( 49,427)
( 50,428)( 51,432)( 52,433)( 53,434)( 54,429)( 55,430)( 56,431)( 57,363)
( 58,364)( 59,365)( 60,369)( 61,370)( 62,371)( 63,366)( 64,367)( 65,368)
( 66,354)( 67,355)( 68,356)( 69,360)( 70,361)( 71,362)( 72,357)( 73,358)
( 74,359)( 75,372)( 76,373)( 77,374)( 78,378)( 79,379)( 80,380)( 81,375)
( 82,376)( 83,377)( 84,336)( 85,337)( 86,338)( 87,342)( 88,343)( 89,344)
( 90,339)( 91,340)( 92,341)( 93,327)( 94,328)( 95,329)( 96,333)( 97,334)
( 98,335)( 99,330)(100,331)(101,332)(102,345)(103,346)(104,347)(105,351)
(106,352)(107,353)(108,348)(109,349)(110,350)(111,228)(112,229)(113,230)
(114,234)(115,235)(116,236)(117,231)(118,232)(119,233)(120,219)(121,220)
(122,221)(123,225)(124,226)(125,227)(126,222)(127,223)(128,224)(129,237)
(130,238)(131,239)(132,243)(133,244)(134,245)(135,240)(136,241)(137,242)
(138,255)(139,256)(140,257)(141,261)(142,262)(143,263)(144,258)(145,259)
(146,260)(147,246)(148,247)(149,248)(150,252)(151,253)(152,254)(153,249)
(154,250)(155,251)(156,264)(157,265)(158,266)(159,270)(160,271)(161,272)
(162,267)(163,268)(164,269)(165,282)(166,283)(167,284)(168,288)(169,289)
(170,290)(171,285)(172,286)(173,287)(174,273)(175,274)(176,275)(177,279)
(178,280)(179,281)(180,276)(181,277)(182,278)(183,291)(184,292)(185,293)
(186,297)(187,298)(188,299)(189,294)(190,295)(191,296)(192,309)(193,310)
(194,311)(195,315)(196,316)(197,317)(198,312)(199,313)(200,314)(201,300)
(202,301)(203,302)(204,306)(205,307)(206,308)(207,303)(208,304)(209,305)
(210,318)(211,319)(212,320)(213,324)(214,325)(215,326)(216,321)(217,322)
(218,323);
poly := sub<Sym(434)|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*s1*s0*s1, s0*s2*s0*s2, s0*s3*s0*s3, 
s1*s3*s1*s3, s2*s3*s2*s3*s2*s3*s2*s3*s2*s3*s2*s3, 
s3*s1*s2*s3*s2*s1*s2*s3*s1*s2*s3*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 >; 
 

to this polytope