Polytope of Type {6,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,8}*864a
if this polytope has a name.
Group : SmallGroup(864,2265)
Rank : 3
Schlafli Type : {6,8}
Number of vertices, edges, etc : 54, 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 :
{6,8,2} of size 1728
Vertex Figure Of :
{2,6,8} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {6,4}*432a
3-fold quotients : {6,8}*288
4-fold quotients : {6,4}*216
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 : {6,16}*1728a, {12,8}*1728a
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 := ( 1,218)( 2,217)( 3,219)( 4,226)( 5,228)( 6,227)( 7,237)( 8,236)
( 9,235)( 10,220)( 11,222)( 12,221)( 13,229)( 14,231)( 15,230)( 16,238)
( 17,240)( 18,239)( 19,225)( 20,224)( 21,223)( 22,232)( 23,234)( 24,233)
( 25,242)( 26,241)( 27,243)( 28,245)( 29,244)( 30,246)( 31,253)( 32,255)
( 33,254)( 34,264)( 35,263)( 36,262)( 37,247)( 38,249)( 39,248)( 40,256)
( 41,258)( 42,257)( 43,265)( 44,267)( 45,266)( 46,252)( 47,251)( 48,250)
( 49,259)( 50,261)( 51,260)( 52,269)( 53,268)( 54,270)( 55,299)( 56,298)
( 57,300)( 58,307)( 59,309)( 60,308)( 61,318)( 62,317)( 63,316)( 64,301)
( 65,303)( 66,302)( 67,310)( 68,312)( 69,311)( 70,319)( 71,321)( 72,320)
( 73,306)( 74,305)( 75,304)( 76,313)( 77,315)( 78,314)( 79,323)( 80,322)
( 81,324)( 82,272)( 83,271)( 84,273)( 85,280)( 86,282)( 87,281)( 88,291)
( 89,290)( 90,289)( 91,274)( 92,276)( 93,275)( 94,283)( 95,285)( 96,284)
( 97,292)( 98,294)( 99,293)(100,279)(101,278)(102,277)(103,286)(104,288)
(105,287)(106,296)(107,295)(108,297)(109,353)(110,352)(111,354)(112,361)
(113,363)(114,362)(115,372)(116,371)(117,370)(118,355)(119,357)(120,356)
(121,364)(122,366)(123,365)(124,373)(125,375)(126,374)(127,360)(128,359)
(129,358)(130,367)(131,369)(132,368)(133,377)(134,376)(135,378)(136,326)
(137,325)(138,327)(139,334)(140,336)(141,335)(142,345)(143,344)(144,343)
(145,328)(146,330)(147,329)(148,337)(149,339)(150,338)(151,346)(152,348)
(153,347)(154,333)(155,332)(156,331)(157,340)(158,342)(159,341)(160,350)
(161,349)(162,351)(163,380)(164,379)(165,381)(166,388)(167,390)(168,389)
(169,399)(170,398)(171,397)(172,382)(173,384)(174,383)(175,391)(176,393)
(177,392)(178,400)(179,402)(180,401)(181,387)(182,386)(183,385)(184,394)
(185,396)(186,395)(187,404)(188,403)(189,405)(190,407)(191,406)(192,408)
(193,415)(194,417)(195,416)(196,426)(197,425)(198,424)(199,409)(200,411)
(201,410)(202,418)(203,420)(204,419)(205,427)(206,429)(207,428)(208,414)
(209,413)(210,412)(211,421)(212,423)(213,422)(214,431)(215,430)(216,432);;
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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)!( 1,218)( 2,217)( 3,219)( 4,226)( 5,228)( 6,227)( 7,237)
( 8,236)( 9,235)( 10,220)( 11,222)( 12,221)( 13,229)( 14,231)( 15,230)
( 16,238)( 17,240)( 18,239)( 19,225)( 20,224)( 21,223)( 22,232)( 23,234)
( 24,233)( 25,242)( 26,241)( 27,243)( 28,245)( 29,244)( 30,246)( 31,253)
( 32,255)( 33,254)( 34,264)( 35,263)( 36,262)( 37,247)( 38,249)( 39,248)
( 40,256)( 41,258)( 42,257)( 43,265)( 44,267)( 45,266)( 46,252)( 47,251)
( 48,250)( 49,259)( 50,261)( 51,260)( 52,269)( 53,268)( 54,270)( 55,299)
( 56,298)( 57,300)( 58,307)( 59,309)( 60,308)( 61,318)( 62,317)( 63,316)
( 64,301)( 65,303)( 66,302)( 67,310)( 68,312)( 69,311)( 70,319)( 71,321)
( 72,320)( 73,306)( 74,305)( 75,304)( 76,313)( 77,315)( 78,314)( 79,323)
( 80,322)( 81,324)( 82,272)( 83,271)( 84,273)( 85,280)( 86,282)( 87,281)
( 88,291)( 89,290)( 90,289)( 91,274)( 92,276)( 93,275)( 94,283)( 95,285)
( 96,284)( 97,292)( 98,294)( 99,293)(100,279)(101,278)(102,277)(103,286)
(104,288)(105,287)(106,296)(107,295)(108,297)(109,353)(110,352)(111,354)
(112,361)(113,363)(114,362)(115,372)(116,371)(117,370)(118,355)(119,357)
(120,356)(121,364)(122,366)(123,365)(124,373)(125,375)(126,374)(127,360)
(128,359)(129,358)(130,367)(131,369)(132,368)(133,377)(134,376)(135,378)
(136,326)(137,325)(138,327)(139,334)(140,336)(141,335)(142,345)(143,344)
(144,343)(145,328)(146,330)(147,329)(148,337)(149,339)(150,338)(151,346)
(152,348)(153,347)(154,333)(155,332)(156,331)(157,340)(158,342)(159,341)
(160,350)(161,349)(162,351)(163,380)(164,379)(165,381)(166,388)(167,390)
(168,389)(169,399)(170,398)(171,397)(172,382)(173,384)(174,383)(175,391)
(176,393)(177,392)(178,400)(179,402)(180,401)(181,387)(182,386)(183,385)
(184,394)(185,396)(186,395)(187,404)(188,403)(189,405)(190,407)(191,406)
(192,408)(193,415)(194,417)(195,416)(196,426)(197,425)(198,424)(199,409)
(200,411)(201,410)(202,418)(203,420)(204,419)(205,427)(206,429)(207,428)
(208,414)(209,413)(210,412)(211,421)(212,423)(213,422)(214,431)(215,430)
(216,432);
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,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
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