Polytope of Type {8,6}

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