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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}*1728h
if this polytope has a name.
Group : SmallGroup(1728,47196)
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}*864i
   3-fold quotients : {2,8,6}*576
   4-fold quotients : {2,12,6}*432c
   6-fold quotients : {2,4,6}*288
   9-fold quotients : {2,24,2}*192
   12-fold quotients : {2,4,6}*144
   18-fold quotients : {2,12,2}*96
   27-fold quotients : {2,8,2}*64
   36-fold quotients : {2,6,2}*48
   54-fold quotients : {2,4,2}*32
   72-fold quotients : {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 := (  4,  5)(  6, 24)(  7, 26)(  8, 25)(  9, 18)( 10, 20)( 11, 19)( 12, 21)
( 13, 23)( 14, 22)( 16, 17)( 28, 29)( 31, 32)( 33, 51)( 34, 53)( 35, 52)
( 36, 45)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 43, 44)( 55, 56)
( 57, 84)( 58, 86)( 59, 85)( 60,105)( 61,107)( 62,106)( 63, 99)( 64,101)
( 65,100)( 66,102)( 67,104)( 68,103)( 69, 96)( 70, 98)( 71, 97)( 72, 90)
( 73, 92)( 74, 91)( 75, 93)( 76, 95)( 77, 94)( 78, 87)( 79, 89)( 80, 88)
( 81,108)( 82,110)( 83,109)(111,138)(112,140)(113,139)(114,159)(115,161)
(116,160)(117,153)(118,155)(119,154)(120,156)(121,158)(122,157)(123,150)
(124,152)(125,151)(126,144)(127,146)(128,145)(129,147)(130,149)(131,148)
(132,141)(133,143)(134,142)(135,162)(136,164)(137,163)(166,167)(168,186)
(169,188)(170,187)(171,180)(172,182)(173,181)(174,183)(175,185)(176,184)
(178,179)(190,191)(193,194)(195,213)(196,215)(197,214)(198,207)(199,209)
(200,208)(201,210)(202,212)(203,211)(205,206)(217,218)(219,273)(220,275)
(221,274)(222,294)(223,296)(224,295)(225,288)(226,290)(227,289)(228,291)
(229,293)(230,292)(231,285)(232,287)(233,286)(234,279)(235,281)(236,280)
(237,282)(238,284)(239,283)(240,276)(241,278)(242,277)(243,297)(244,299)
(245,298)(246,300)(247,302)(248,301)(249,321)(250,323)(251,322)(252,315)
(253,317)(254,316)(255,318)(256,320)(257,319)(258,312)(259,314)(260,313)
(261,306)(262,308)(263,307)(264,309)(265,311)(266,310)(267,303)(268,305)
(269,304)(270,324)(271,326)(272,325)(327,408)(328,410)(329,409)(330,429)
(331,431)(332,430)(333,423)(334,425)(335,424)(336,426)(337,428)(338,427)
(339,420)(340,422)(341,421)(342,414)(343,416)(344,415)(345,417)(346,419)
(347,418)(348,411)(349,413)(350,412)(351,432)(352,434)(353,433)(354,381)
(355,383)(356,382)(357,402)(358,404)(359,403)(360,396)(361,398)(362,397)
(363,399)(364,401)(365,400)(366,393)(367,395)(368,394)(369,387)(370,389)
(371,388)(372,390)(373,392)(374,391)(375,384)(376,386)(377,385)(378,405)
(379,407)(380,406);;
s2 := (  3,220)(  4,219)(  5,221)(  6,226)(  7,225)(  8,227)(  9,223)( 10,222)
( 11,224)( 12,232)( 13,231)( 14,233)( 15,229)( 16,228)( 17,230)( 18,235)
( 19,234)( 20,236)( 21,244)( 22,243)( 23,245)( 24,241)( 25,240)( 26,242)
( 27,238)( 28,237)( 29,239)( 30,247)( 31,246)( 32,248)( 33,253)( 34,252)
( 35,254)( 36,250)( 37,249)( 38,251)( 39,259)( 40,258)( 41,260)( 42,256)
( 43,255)( 44,257)( 45,262)( 46,261)( 47,263)( 48,271)( 49,270)( 50,272)
( 51,268)( 52,267)( 53,269)( 54,265)( 55,264)( 56,266)( 57,301)( 58,300)
( 59,302)( 60,307)( 61,306)( 62,308)( 63,304)( 64,303)( 65,305)( 66,313)
( 67,312)( 68,314)( 69,310)( 70,309)( 71,311)( 72,316)( 73,315)( 74,317)
( 75,325)( 76,324)( 77,326)( 78,322)( 79,321)( 80,323)( 81,319)( 82,318)
( 83,320)( 84,274)( 85,273)( 86,275)( 87,280)( 88,279)( 89,281)( 90,277)
( 91,276)( 92,278)( 93,286)( 94,285)( 95,287)( 96,283)( 97,282)( 98,284)
( 99,289)(100,288)(101,290)(102,298)(103,297)(104,299)(105,295)(106,294)
(107,296)(108,292)(109,291)(110,293)(111,355)(112,354)(113,356)(114,361)
(115,360)(116,362)(117,358)(118,357)(119,359)(120,367)(121,366)(122,368)
(123,364)(124,363)(125,365)(126,370)(127,369)(128,371)(129,379)(130,378)
(131,380)(132,376)(133,375)(134,377)(135,373)(136,372)(137,374)(138,328)
(139,327)(140,329)(141,334)(142,333)(143,335)(144,331)(145,330)(146,332)
(147,340)(148,339)(149,341)(150,337)(151,336)(152,338)(153,343)(154,342)
(155,344)(156,352)(157,351)(158,353)(159,349)(160,348)(161,350)(162,346)
(163,345)(164,347)(165,382)(166,381)(167,383)(168,388)(169,387)(170,389)
(171,385)(172,384)(173,386)(174,394)(175,393)(176,395)(177,391)(178,390)
(179,392)(180,397)(181,396)(182,398)(183,406)(184,405)(185,407)(186,403)
(187,402)(188,404)(189,400)(190,399)(191,401)(192,409)(193,408)(194,410)
(195,415)(196,414)(197,416)(198,412)(199,411)(200,413)(201,421)(202,420)
(203,422)(204,418)(205,417)(206,419)(207,424)(208,423)(209,425)(210,433)
(211,432)(212,434)(213,430)(214,429)(215,431)(216,427)(217,426)(218,428);;
s3 := (  3,177)(  4,178)(  5,179)(  6,174)(  7,175)(  8,176)(  9,180)( 10,181)
( 11,182)( 12,168)( 13,169)( 14,170)( 15,165)( 16,166)( 17,167)( 18,171)
( 19,172)( 20,173)( 21,186)( 22,187)( 23,188)( 24,183)( 25,184)( 26,185)
( 27,189)( 28,190)( 29,191)( 30,204)( 31,205)( 32,206)( 33,201)( 34,202)
( 35,203)( 36,207)( 37,208)( 38,209)( 39,195)( 40,196)( 41,197)( 42,192)
( 43,193)( 44,194)( 45,198)( 46,199)( 47,200)( 48,213)( 49,214)( 50,215)
( 51,210)( 52,211)( 53,212)( 54,216)( 55,217)( 56,218)( 57,150)( 58,151)
( 59,152)( 60,147)( 61,148)( 62,149)( 63,153)( 64,154)( 65,155)( 66,141)
( 67,142)( 68,143)( 69,138)( 70,139)( 71,140)( 72,144)( 73,145)( 74,146)
( 75,159)( 76,160)( 77,161)( 78,156)( 79,157)( 80,158)( 81,162)( 82,163)
( 83,164)( 84,123)( 85,124)( 86,125)( 87,120)( 88,121)( 89,122)( 90,126)
( 91,127)( 92,128)( 93,114)( 94,115)( 95,116)( 96,111)( 97,112)( 98,113)
( 99,117)(100,118)(101,119)(102,132)(103,133)(104,134)(105,129)(106,130)
(107,131)(108,135)(109,136)(110,137)(219,393)(220,394)(221,395)(222,390)
(223,391)(224,392)(225,396)(226,397)(227,398)(228,384)(229,385)(230,386)
(231,381)(232,382)(233,383)(234,387)(235,388)(236,389)(237,402)(238,403)
(239,404)(240,399)(241,400)(242,401)(243,405)(244,406)(245,407)(246,420)
(247,421)(248,422)(249,417)(250,418)(251,419)(252,423)(253,424)(254,425)
(255,411)(256,412)(257,413)(258,408)(259,409)(260,410)(261,414)(262,415)
(263,416)(264,429)(265,430)(266,431)(267,426)(268,427)(269,428)(270,432)
(271,433)(272,434)(273,366)(274,367)(275,368)(276,363)(277,364)(278,365)
(279,369)(280,370)(281,371)(282,357)(283,358)(284,359)(285,354)(286,355)
(287,356)(288,360)(289,361)(290,362)(291,375)(292,376)(293,377)(294,372)
(295,373)(296,374)(297,378)(298,379)(299,380)(300,339)(301,340)(302,341)
(303,336)(304,337)(305,338)(306,342)(307,343)(308,344)(309,330)(310,331)
(311,332)(312,327)(313,328)(314,329)(315,333)(316,334)(317,335)(318,348)
(319,349)(320,350)(321,345)(322,346)(323,347)(324,351)(325,352)(326,353);;
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, 
s1*s2*s3*s2*s3*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)!(  4,  5)(  6, 24)(  7, 26)(  8, 25)(  9, 18)( 10, 20)( 11, 19)
( 12, 21)( 13, 23)( 14, 22)( 16, 17)( 28, 29)( 31, 32)( 33, 51)( 34, 53)
( 35, 52)( 36, 45)( 37, 47)( 38, 46)( 39, 48)( 40, 50)( 41, 49)( 43, 44)
( 55, 56)( 57, 84)( 58, 86)( 59, 85)( 60,105)( 61,107)( 62,106)( 63, 99)
( 64,101)( 65,100)( 66,102)( 67,104)( 68,103)( 69, 96)( 70, 98)( 71, 97)
( 72, 90)( 73, 92)( 74, 91)( 75, 93)( 76, 95)( 77, 94)( 78, 87)( 79, 89)
( 80, 88)( 81,108)( 82,110)( 83,109)(111,138)(112,140)(113,139)(114,159)
(115,161)(116,160)(117,153)(118,155)(119,154)(120,156)(121,158)(122,157)
(123,150)(124,152)(125,151)(126,144)(127,146)(128,145)(129,147)(130,149)
(131,148)(132,141)(133,143)(134,142)(135,162)(136,164)(137,163)(166,167)
(168,186)(169,188)(170,187)(171,180)(172,182)(173,181)(174,183)(175,185)
(176,184)(178,179)(190,191)(193,194)(195,213)(196,215)(197,214)(198,207)
(199,209)(200,208)(201,210)(202,212)(203,211)(205,206)(217,218)(219,273)
(220,275)(221,274)(222,294)(223,296)(224,295)(225,288)(226,290)(227,289)
(228,291)(229,293)(230,292)(231,285)(232,287)(233,286)(234,279)(235,281)
(236,280)(237,282)(238,284)(239,283)(240,276)(241,278)(242,277)(243,297)
(244,299)(245,298)(246,300)(247,302)(248,301)(249,321)(250,323)(251,322)
(252,315)(253,317)(254,316)(255,318)(256,320)(257,319)(258,312)(259,314)
(260,313)(261,306)(262,308)(263,307)(264,309)(265,311)(266,310)(267,303)
(268,305)(269,304)(270,324)(271,326)(272,325)(327,408)(328,410)(329,409)
(330,429)(331,431)(332,430)(333,423)(334,425)(335,424)(336,426)(337,428)
(338,427)(339,420)(340,422)(341,421)(342,414)(343,416)(344,415)(345,417)
(346,419)(347,418)(348,411)(349,413)(350,412)(351,432)(352,434)(353,433)
(354,381)(355,383)(356,382)(357,402)(358,404)(359,403)(360,396)(361,398)
(362,397)(363,399)(364,401)(365,400)(366,393)(367,395)(368,394)(369,387)
(370,389)(371,388)(372,390)(373,392)(374,391)(375,384)(376,386)(377,385)
(378,405)(379,407)(380,406);
s2 := Sym(434)!(  3,220)(  4,219)(  5,221)(  6,226)(  7,225)(  8,227)(  9,223)
( 10,222)( 11,224)( 12,232)( 13,231)( 14,233)( 15,229)( 16,228)( 17,230)
( 18,235)( 19,234)( 20,236)( 21,244)( 22,243)( 23,245)( 24,241)( 25,240)
( 26,242)( 27,238)( 28,237)( 29,239)( 30,247)( 31,246)( 32,248)( 33,253)
( 34,252)( 35,254)( 36,250)( 37,249)( 38,251)( 39,259)( 40,258)( 41,260)
( 42,256)( 43,255)( 44,257)( 45,262)( 46,261)( 47,263)( 48,271)( 49,270)
( 50,272)( 51,268)( 52,267)( 53,269)( 54,265)( 55,264)( 56,266)( 57,301)
( 58,300)( 59,302)( 60,307)( 61,306)( 62,308)( 63,304)( 64,303)( 65,305)
( 66,313)( 67,312)( 68,314)( 69,310)( 70,309)( 71,311)( 72,316)( 73,315)
( 74,317)( 75,325)( 76,324)( 77,326)( 78,322)( 79,321)( 80,323)( 81,319)
( 82,318)( 83,320)( 84,274)( 85,273)( 86,275)( 87,280)( 88,279)( 89,281)
( 90,277)( 91,276)( 92,278)( 93,286)( 94,285)( 95,287)( 96,283)( 97,282)
( 98,284)( 99,289)(100,288)(101,290)(102,298)(103,297)(104,299)(105,295)
(106,294)(107,296)(108,292)(109,291)(110,293)(111,355)(112,354)(113,356)
(114,361)(115,360)(116,362)(117,358)(118,357)(119,359)(120,367)(121,366)
(122,368)(123,364)(124,363)(125,365)(126,370)(127,369)(128,371)(129,379)
(130,378)(131,380)(132,376)(133,375)(134,377)(135,373)(136,372)(137,374)
(138,328)(139,327)(140,329)(141,334)(142,333)(143,335)(144,331)(145,330)
(146,332)(147,340)(148,339)(149,341)(150,337)(151,336)(152,338)(153,343)
(154,342)(155,344)(156,352)(157,351)(158,353)(159,349)(160,348)(161,350)
(162,346)(163,345)(164,347)(165,382)(166,381)(167,383)(168,388)(169,387)
(170,389)(171,385)(172,384)(173,386)(174,394)(175,393)(176,395)(177,391)
(178,390)(179,392)(180,397)(181,396)(182,398)(183,406)(184,405)(185,407)
(186,403)(187,402)(188,404)(189,400)(190,399)(191,401)(192,409)(193,408)
(194,410)(195,415)(196,414)(197,416)(198,412)(199,411)(200,413)(201,421)
(202,420)(203,422)(204,418)(205,417)(206,419)(207,424)(208,423)(209,425)
(210,433)(211,432)(212,434)(213,430)(214,429)(215,431)(216,427)(217,426)
(218,428);
s3 := Sym(434)!(  3,177)(  4,178)(  5,179)(  6,174)(  7,175)(  8,176)(  9,180)
( 10,181)( 11,182)( 12,168)( 13,169)( 14,170)( 15,165)( 16,166)( 17,167)
( 18,171)( 19,172)( 20,173)( 21,186)( 22,187)( 23,188)( 24,183)( 25,184)
( 26,185)( 27,189)( 28,190)( 29,191)( 30,204)( 31,205)( 32,206)( 33,201)
( 34,202)( 35,203)( 36,207)( 37,208)( 38,209)( 39,195)( 40,196)( 41,197)
( 42,192)( 43,193)( 44,194)( 45,198)( 46,199)( 47,200)( 48,213)( 49,214)
( 50,215)( 51,210)( 52,211)( 53,212)( 54,216)( 55,217)( 56,218)( 57,150)
( 58,151)( 59,152)( 60,147)( 61,148)( 62,149)( 63,153)( 64,154)( 65,155)
( 66,141)( 67,142)( 68,143)( 69,138)( 70,139)( 71,140)( 72,144)( 73,145)
( 74,146)( 75,159)( 76,160)( 77,161)( 78,156)( 79,157)( 80,158)( 81,162)
( 82,163)( 83,164)( 84,123)( 85,124)( 86,125)( 87,120)( 88,121)( 89,122)
( 90,126)( 91,127)( 92,128)( 93,114)( 94,115)( 95,116)( 96,111)( 97,112)
( 98,113)( 99,117)(100,118)(101,119)(102,132)(103,133)(104,134)(105,129)
(106,130)(107,131)(108,135)(109,136)(110,137)(219,393)(220,394)(221,395)
(222,390)(223,391)(224,392)(225,396)(226,397)(227,398)(228,384)(229,385)
(230,386)(231,381)(232,382)(233,383)(234,387)(235,388)(236,389)(237,402)
(238,403)(239,404)(240,399)(241,400)(242,401)(243,405)(244,406)(245,407)
(246,420)(247,421)(248,422)(249,417)(250,418)(251,419)(252,423)(253,424)
(254,425)(255,411)(256,412)(257,413)(258,408)(259,409)(260,410)(261,414)
(262,415)(263,416)(264,429)(265,430)(266,431)(267,426)(268,427)(269,428)
(270,432)(271,433)(272,434)(273,366)(274,367)(275,368)(276,363)(277,364)
(278,365)(279,369)(280,370)(281,371)(282,357)(283,358)(284,359)(285,354)
(286,355)(287,356)(288,360)(289,361)(290,362)(291,375)(292,376)(293,377)
(294,372)(295,373)(296,374)(297,378)(298,379)(299,380)(300,339)(301,340)
(302,341)(303,336)(304,337)(305,338)(306,342)(307,343)(308,344)(309,330)
(310,331)(311,332)(312,327)(313,328)(314,329)(315,333)(316,334)(317,335)
(318,348)(319,349)(320,350)(321,345)(322,346)(323,347)(324,351)(325,352)
(326,353);
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, 
s1*s2*s3*s2*s3*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