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