Polytope of Type {24,6}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {24,6}*864h
if this polytope has a name.
Group : SmallGroup(864,4303)
Rank : 3
Schlafli Type : {24,6}
Number of vertices, edges, etc : 72, 216, 18
Order of s₀s₁s₂ : 24
Order of s₀s₁s₂s₁ : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Self-Petrie
Related Polytopes :
   Facet
   Vertex Figure
   Dual
   Petrial
Facet Of :
   {24,6,2} of size 1728
Vertex Figure Of :
   {2,24,6} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {12,6}*432i
   3-fold quotients : {8,6}*288
   4-fold quotients : {12,6}*216c
   6-fold quotients : {4,6}*144
   9-fold quotients : {24,2}*96
   12-fold quotients : {4,6}*72
   18-fold quotients : {12,2}*48
   27-fold quotients : {8,2}*32
   36-fold quotients : {6,2}*24
   54-fold quotients : {4,2}*16
   72-fold quotients : {3,2}*12
   108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
   2-fold covers : {48,6}*1728h, {24,12}*1728u
Permutation Representation (GAP) :

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