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Polytope of Type {6,24}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,24}*864e
if this polytope has a name.
Group : SmallGroup(864,2265)
Rank : 3
Schlafli Type : {6,24}
Number of vertices, edges, etc : 18, 216, 72
Order of s0s1s2 : 8
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {6,24,2} of size 1728
Vertex Figure Of :
   {2,6,24} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {6,12}*432f
   3-fold quotients : {6,8}*288
   4-fold quotients : {6,12}*216b
   6-fold quotients : {6,4}*144
   12-fold quotients : {6,4}*72
   27-fold quotients : {2,8}*32
   54-fold quotients : {2,4}*16
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {6,48}*1728e, {12,24}*1728g
Permutation Representation (GAP) :
s0 := (  1,163)(  2,164)(  3,165)(  4,171)(  5,169)(  6,170)(  7,167)(  8,168)
(  9,166)( 10,181)( 11,182)( 12,183)( 13,189)( 14,187)( 15,188)( 16,185)
( 17,186)( 18,184)( 19,172)( 20,173)( 21,174)( 22,180)( 23,178)( 24,179)
( 25,176)( 26,177)( 27,175)( 28,190)( 29,191)( 30,192)( 31,198)( 32,196)
( 33,197)( 34,194)( 35,195)( 36,193)( 37,208)( 38,209)( 39,210)( 40,216)
( 41,214)( 42,215)( 43,212)( 44,213)( 45,211)( 46,199)( 47,200)( 48,201)
( 49,207)( 50,205)( 51,206)( 52,203)( 53,204)( 54,202)( 55,136)( 56,137)
( 57,138)( 58,144)( 59,142)( 60,143)( 61,140)( 62,141)( 63,139)( 64,154)
( 65,155)( 66,156)( 67,162)( 68,160)( 69,161)( 70,158)( 71,159)( 72,157)
( 73,145)( 74,146)( 75,147)( 76,153)( 77,151)( 78,152)( 79,149)( 80,150)
( 81,148)( 82,109)( 83,110)( 84,111)( 85,117)( 86,115)( 87,116)( 88,113)
( 89,114)( 90,112)( 91,127)( 92,128)( 93,129)( 94,135)( 95,133)( 96,134)
( 97,131)( 98,132)( 99,130)(100,118)(101,119)(102,120)(103,126)(104,124)
(105,125)(106,122)(107,123)(108,121)(217,379)(218,380)(219,381)(220,387)
(221,385)(222,386)(223,383)(224,384)(225,382)(226,397)(227,398)(228,399)
(229,405)(230,403)(231,404)(232,401)(233,402)(234,400)(235,388)(236,389)
(237,390)(238,396)(239,394)(240,395)(241,392)(242,393)(243,391)(244,406)
(245,407)(246,408)(247,414)(248,412)(249,413)(250,410)(251,411)(252,409)
(253,424)(254,425)(255,426)(256,432)(257,430)(258,431)(259,428)(260,429)
(261,427)(262,415)(263,416)(264,417)(265,423)(266,421)(267,422)(268,419)
(269,420)(270,418)(271,352)(272,353)(273,354)(274,360)(275,358)(276,359)
(277,356)(278,357)(279,355)(280,370)(281,371)(282,372)(283,378)(284,376)
(285,377)(286,374)(287,375)(288,373)(289,361)(290,362)(291,363)(292,369)
(293,367)(294,368)(295,365)(296,366)(297,364)(298,325)(299,326)(300,327)
(301,333)(302,331)(303,332)(304,329)(305,330)(306,328)(307,343)(308,344)
(309,345)(310,351)(311,349)(312,350)(313,347)(314,348)(315,346)(316,334)
(317,335)(318,336)(319,342)(320,340)(321,341)(322,338)(323,339)(324,337);;
s1 := (  1, 10)(  2, 12)(  3, 11)(  4, 15)(  5, 14)(  6, 13)(  7, 17)(  8, 16)
(  9, 18)( 20, 21)( 22, 24)( 25, 26)( 28, 37)( 29, 39)( 30, 38)( 31, 42)
( 32, 41)( 33, 40)( 34, 44)( 35, 43)( 36, 45)( 47, 48)( 49, 51)( 52, 53)
( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)( 62, 97)
( 63, 99)( 64, 82)( 65, 84)( 66, 83)( 67, 87)( 68, 86)( 69, 85)( 70, 89)
( 71, 88)( 72, 90)( 73,100)( 74,102)( 75,101)( 76,105)( 77,104)( 78,103)
( 79,107)( 80,106)( 81,108)(109,145)(110,147)(111,146)(112,150)(113,149)
(114,148)(115,152)(116,151)(117,153)(118,136)(119,138)(120,137)(121,141)
(122,140)(123,139)(124,143)(125,142)(126,144)(127,154)(128,156)(129,155)
(130,159)(131,158)(132,157)(133,161)(134,160)(135,162)(163,172)(164,174)
(165,173)(166,177)(167,176)(168,175)(169,179)(170,178)(171,180)(182,183)
(184,186)(187,188)(190,199)(191,201)(192,200)(193,204)(194,203)(195,202)
(196,206)(197,205)(198,207)(209,210)(211,213)(214,215)(217,280)(218,282)
(219,281)(220,285)(221,284)(222,283)(223,287)(224,286)(225,288)(226,271)
(227,273)(228,272)(229,276)(230,275)(231,274)(232,278)(233,277)(234,279)
(235,289)(236,291)(237,290)(238,294)(239,293)(240,292)(241,296)(242,295)
(243,297)(244,307)(245,309)(246,308)(247,312)(248,311)(249,310)(250,314)
(251,313)(252,315)(253,298)(254,300)(255,299)(256,303)(257,302)(258,301)
(259,305)(260,304)(261,306)(262,316)(263,318)(264,317)(265,321)(266,320)
(267,319)(268,323)(269,322)(270,324)(325,415)(326,417)(327,416)(328,420)
(329,419)(330,418)(331,422)(332,421)(333,423)(334,406)(335,408)(336,407)
(337,411)(338,410)(339,409)(340,413)(341,412)(342,414)(343,424)(344,426)
(345,425)(346,429)(347,428)(348,427)(349,431)(350,430)(351,432)(352,388)
(353,390)(354,389)(355,393)(356,392)(357,391)(358,395)(359,394)(360,396)
(361,379)(362,381)(363,380)(364,384)(365,383)(366,382)(367,386)(368,385)
(369,387)(370,397)(371,399)(372,398)(373,402)(374,401)(375,400)(376,404)
(377,403)(378,405);;
s2 := (  1,272)(  2,271)(  3,273)(  4,289)(  5,291)(  6,290)(  7,282)(  8,281)
(  9,280)( 10,279)( 11,278)( 12,277)( 13,297)( 14,296)( 15,295)( 16,288)
( 17,287)( 18,286)( 19,274)( 20,276)( 21,275)( 22,293)( 23,292)( 24,294)
( 25,285)( 26,284)( 27,283)( 28,299)( 29,298)( 30,300)( 31,316)( 32,318)
( 33,317)( 34,309)( 35,308)( 36,307)( 37,306)( 38,305)( 39,304)( 40,324)
( 41,323)( 42,322)( 43,315)( 44,314)( 45,313)( 46,301)( 47,303)( 48,302)
( 49,320)( 50,319)( 51,321)( 52,312)( 53,311)( 54,310)( 55,218)( 56,217)
( 57,219)( 58,235)( 59,237)( 60,236)( 61,228)( 62,227)( 63,226)( 64,225)
( 65,224)( 66,223)( 67,243)( 68,242)( 69,241)( 70,234)( 71,233)( 72,232)
( 73,220)( 74,222)( 75,221)( 76,239)( 77,238)( 78,240)( 79,231)( 80,230)
( 81,229)( 82,245)( 83,244)( 84,246)( 85,262)( 86,264)( 87,263)( 88,255)
( 89,254)( 90,253)( 91,252)( 92,251)( 93,250)( 94,270)( 95,269)( 96,268)
( 97,261)( 98,260)( 99,259)(100,247)(101,249)(102,248)(103,266)(104,265)
(105,267)(106,258)(107,257)(108,256)(109,407)(110,406)(111,408)(112,424)
(113,426)(114,425)(115,417)(116,416)(117,415)(118,414)(119,413)(120,412)
(121,432)(122,431)(123,430)(124,423)(125,422)(126,421)(127,409)(128,411)
(129,410)(130,428)(131,427)(132,429)(133,420)(134,419)(135,418)(136,380)
(137,379)(138,381)(139,397)(140,399)(141,398)(142,390)(143,389)(144,388)
(145,387)(146,386)(147,385)(148,405)(149,404)(150,403)(151,396)(152,395)
(153,394)(154,382)(155,384)(156,383)(157,401)(158,400)(159,402)(160,393)
(161,392)(162,391)(163,353)(164,352)(165,354)(166,370)(167,372)(168,371)
(169,363)(170,362)(171,361)(172,360)(173,359)(174,358)(175,378)(176,377)
(177,376)(178,369)(179,368)(180,367)(181,355)(182,357)(183,356)(184,374)
(185,373)(186,375)(187,366)(188,365)(189,364)(190,326)(191,325)(192,327)
(193,343)(194,345)(195,344)(196,336)(197,335)(198,334)(199,333)(200,332)
(201,331)(202,351)(203,350)(204,349)(205,342)(206,341)(207,340)(208,328)
(209,330)(210,329)(211,347)(212,346)(213,348)(214,339)(215,338)(216,337);;
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, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(432)!(  1,163)(  2,164)(  3,165)(  4,171)(  5,169)(  6,170)(  7,167)
(  8,168)(  9,166)( 10,181)( 11,182)( 12,183)( 13,189)( 14,187)( 15,188)
( 16,185)( 17,186)( 18,184)( 19,172)( 20,173)( 21,174)( 22,180)( 23,178)
( 24,179)( 25,176)( 26,177)( 27,175)( 28,190)( 29,191)( 30,192)( 31,198)
( 32,196)( 33,197)( 34,194)( 35,195)( 36,193)( 37,208)( 38,209)( 39,210)
( 40,216)( 41,214)( 42,215)( 43,212)( 44,213)( 45,211)( 46,199)( 47,200)
( 48,201)( 49,207)( 50,205)( 51,206)( 52,203)( 53,204)( 54,202)( 55,136)
( 56,137)( 57,138)( 58,144)( 59,142)( 60,143)( 61,140)( 62,141)( 63,139)
( 64,154)( 65,155)( 66,156)( 67,162)( 68,160)( 69,161)( 70,158)( 71,159)
( 72,157)( 73,145)( 74,146)( 75,147)( 76,153)( 77,151)( 78,152)( 79,149)
( 80,150)( 81,148)( 82,109)( 83,110)( 84,111)( 85,117)( 86,115)( 87,116)
( 88,113)( 89,114)( 90,112)( 91,127)( 92,128)( 93,129)( 94,135)( 95,133)
( 96,134)( 97,131)( 98,132)( 99,130)(100,118)(101,119)(102,120)(103,126)
(104,124)(105,125)(106,122)(107,123)(108,121)(217,379)(218,380)(219,381)
(220,387)(221,385)(222,386)(223,383)(224,384)(225,382)(226,397)(227,398)
(228,399)(229,405)(230,403)(231,404)(232,401)(233,402)(234,400)(235,388)
(236,389)(237,390)(238,396)(239,394)(240,395)(241,392)(242,393)(243,391)
(244,406)(245,407)(246,408)(247,414)(248,412)(249,413)(250,410)(251,411)
(252,409)(253,424)(254,425)(255,426)(256,432)(257,430)(258,431)(259,428)
(260,429)(261,427)(262,415)(263,416)(264,417)(265,423)(266,421)(267,422)
(268,419)(269,420)(270,418)(271,352)(272,353)(273,354)(274,360)(275,358)
(276,359)(277,356)(278,357)(279,355)(280,370)(281,371)(282,372)(283,378)
(284,376)(285,377)(286,374)(287,375)(288,373)(289,361)(290,362)(291,363)
(292,369)(293,367)(294,368)(295,365)(296,366)(297,364)(298,325)(299,326)
(300,327)(301,333)(302,331)(303,332)(304,329)(305,330)(306,328)(307,343)
(308,344)(309,345)(310,351)(311,349)(312,350)(313,347)(314,348)(315,346)
(316,334)(317,335)(318,336)(319,342)(320,340)(321,341)(322,338)(323,339)
(324,337);
s1 := Sym(432)!(  1, 10)(  2, 12)(  3, 11)(  4, 15)(  5, 14)(  6, 13)(  7, 17)
(  8, 16)(  9, 18)( 20, 21)( 22, 24)( 25, 26)( 28, 37)( 29, 39)( 30, 38)
( 31, 42)( 32, 41)( 33, 40)( 34, 44)( 35, 43)( 36, 45)( 47, 48)( 49, 51)
( 52, 53)( 55, 91)( 56, 93)( 57, 92)( 58, 96)( 59, 95)( 60, 94)( 61, 98)
( 62, 97)( 63, 99)( 64, 82)( 65, 84)( 66, 83)( 67, 87)( 68, 86)( 69, 85)
( 70, 89)( 71, 88)( 72, 90)( 73,100)( 74,102)( 75,101)( 76,105)( 77,104)
( 78,103)( 79,107)( 80,106)( 81,108)(109,145)(110,147)(111,146)(112,150)
(113,149)(114,148)(115,152)(116,151)(117,153)(118,136)(119,138)(120,137)
(121,141)(122,140)(123,139)(124,143)(125,142)(126,144)(127,154)(128,156)
(129,155)(130,159)(131,158)(132,157)(133,161)(134,160)(135,162)(163,172)
(164,174)(165,173)(166,177)(167,176)(168,175)(169,179)(170,178)(171,180)
(182,183)(184,186)(187,188)(190,199)(191,201)(192,200)(193,204)(194,203)
(195,202)(196,206)(197,205)(198,207)(209,210)(211,213)(214,215)(217,280)
(218,282)(219,281)(220,285)(221,284)(222,283)(223,287)(224,286)(225,288)
(226,271)(227,273)(228,272)(229,276)(230,275)(231,274)(232,278)(233,277)
(234,279)(235,289)(236,291)(237,290)(238,294)(239,293)(240,292)(241,296)
(242,295)(243,297)(244,307)(245,309)(246,308)(247,312)(248,311)(249,310)
(250,314)(251,313)(252,315)(253,298)(254,300)(255,299)(256,303)(257,302)
(258,301)(259,305)(260,304)(261,306)(262,316)(263,318)(264,317)(265,321)
(266,320)(267,319)(268,323)(269,322)(270,324)(325,415)(326,417)(327,416)
(328,420)(329,419)(330,418)(331,422)(332,421)(333,423)(334,406)(335,408)
(336,407)(337,411)(338,410)(339,409)(340,413)(341,412)(342,414)(343,424)
(344,426)(345,425)(346,429)(347,428)(348,427)(349,431)(350,430)(351,432)
(352,388)(353,390)(354,389)(355,393)(356,392)(357,391)(358,395)(359,394)
(360,396)(361,379)(362,381)(363,380)(364,384)(365,383)(366,382)(367,386)
(368,385)(369,387)(370,397)(371,399)(372,398)(373,402)(374,401)(375,400)
(376,404)(377,403)(378,405);
s2 := Sym(432)!(  1,272)(  2,271)(  3,273)(  4,289)(  5,291)(  6,290)(  7,282)
(  8,281)(  9,280)( 10,279)( 11,278)( 12,277)( 13,297)( 14,296)( 15,295)
( 16,288)( 17,287)( 18,286)( 19,274)( 20,276)( 21,275)( 22,293)( 23,292)
( 24,294)( 25,285)( 26,284)( 27,283)( 28,299)( 29,298)( 30,300)( 31,316)
( 32,318)( 33,317)( 34,309)( 35,308)( 36,307)( 37,306)( 38,305)( 39,304)
( 40,324)( 41,323)( 42,322)( 43,315)( 44,314)( 45,313)( 46,301)( 47,303)
( 48,302)( 49,320)( 50,319)( 51,321)( 52,312)( 53,311)( 54,310)( 55,218)
( 56,217)( 57,219)( 58,235)( 59,237)( 60,236)( 61,228)( 62,227)( 63,226)
( 64,225)( 65,224)( 66,223)( 67,243)( 68,242)( 69,241)( 70,234)( 71,233)
( 72,232)( 73,220)( 74,222)( 75,221)( 76,239)( 77,238)( 78,240)( 79,231)
( 80,230)( 81,229)( 82,245)( 83,244)( 84,246)( 85,262)( 86,264)( 87,263)
( 88,255)( 89,254)( 90,253)( 91,252)( 92,251)( 93,250)( 94,270)( 95,269)
( 96,268)( 97,261)( 98,260)( 99,259)(100,247)(101,249)(102,248)(103,266)
(104,265)(105,267)(106,258)(107,257)(108,256)(109,407)(110,406)(111,408)
(112,424)(113,426)(114,425)(115,417)(116,416)(117,415)(118,414)(119,413)
(120,412)(121,432)(122,431)(123,430)(124,423)(125,422)(126,421)(127,409)
(128,411)(129,410)(130,428)(131,427)(132,429)(133,420)(134,419)(135,418)
(136,380)(137,379)(138,381)(139,397)(140,399)(141,398)(142,390)(143,389)
(144,388)(145,387)(146,386)(147,385)(148,405)(149,404)(150,403)(151,396)
(152,395)(153,394)(154,382)(155,384)(156,383)(157,401)(158,400)(159,402)
(160,393)(161,392)(162,391)(163,353)(164,352)(165,354)(166,370)(167,372)
(168,371)(169,363)(170,362)(171,361)(172,360)(173,359)(174,358)(175,378)
(176,377)(177,376)(178,369)(179,368)(180,367)(181,355)(182,357)(183,356)
(184,374)(185,373)(186,375)(187,366)(188,365)(189,364)(190,326)(191,325)
(192,327)(193,343)(194,345)(195,344)(196,336)(197,335)(198,334)(199,333)
(200,332)(201,331)(202,351)(203,350)(204,349)(205,342)(206,341)(207,340)
(208,328)(209,330)(210,329)(211,347)(212,346)(213,348)(214,339)(215,338)
(216,337);
poly := sub<Sym(432)|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, 
s0*s1*s2*s1*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1*s2*s1, 
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1, 
s0*s1*s2*s1*s2*s1*s0*s1*s0*s1*s2*s1*s2*s1*s2*s0*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope