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Polytope of Type {8,27}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,27}*864
if this polytope has a name.
Group : SmallGroup(864,619)
Rank : 3
Schlafli Type : {8,27}
Number of vertices, edges, etc : 16, 216, 54
Order of s0s1s2 : 108
Order of s0s1s2s1 : 8
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
Related Polytopes :
   Facet
   Vertex Figure
   Dual
Facet Of :
   {8,27,2} of size 1728
Vertex Figure Of :
   {2,8,27} of size 1728
Quotients (Maximal Quotients in Boldface) :
   2-fold quotients : {4,27}*432
   3-fold quotients : {8,9}*288
   4-fold quotients : {4,27}*216
   6-fold quotients : {4,9}*144
   8-fold quotients : {2,27}*108
   9-fold quotients : {8,3}*96
   12-fold quotients : {4,9}*72
   18-fold quotients : {4,3}*48
   24-fold quotients : {2,9}*36
   36-fold quotients : {4,3}*24
   72-fold quotients : {2,3}*12
Covers (Minimal Covers in Boldface) :
   2-fold covers : {8,54}*1728b
Permutation Representation (GAP) :
s0 := (  1,221)(  2,222)(  3,224)(  4,223)(  5,218)(  6,217)(  7,219)(  8,220)
(  9,229)( 10,230)( 11,232)( 12,231)( 13,226)( 14,225)( 15,227)( 16,228)
( 17,237)( 18,238)( 19,240)( 20,239)( 21,234)( 22,233)( 23,235)( 24,236)
( 25,245)( 26,246)( 27,248)( 28,247)( 29,242)( 30,241)( 31,243)( 32,244)
( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)( 40,252)
( 41,261)( 42,262)( 43,264)( 44,263)( 45,258)( 46,257)( 47,259)( 48,260)
( 49,269)( 50,270)( 51,272)( 52,271)( 53,266)( 54,265)( 55,267)( 56,268)
( 57,277)( 58,278)( 59,280)( 60,279)( 61,274)( 62,273)( 63,275)( 64,276)
( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)( 72,284)
( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)( 80,292)
( 81,301)( 82,302)( 83,304)( 84,303)( 85,298)( 86,297)( 87,299)( 88,300)
( 89,309)( 90,310)( 91,312)( 92,311)( 93,306)( 94,305)( 95,307)( 96,308)
( 97,317)( 98,318)( 99,320)(100,319)(101,314)(102,313)(103,315)(104,316)
(105,325)(106,326)(107,328)(108,327)(109,322)(110,321)(111,323)(112,324)
(113,333)(114,334)(115,336)(116,335)(117,330)(118,329)(119,331)(120,332)
(121,341)(122,342)(123,344)(124,343)(125,338)(126,337)(127,339)(128,340)
(129,349)(130,350)(131,352)(132,351)(133,346)(134,345)(135,347)(136,348)
(137,357)(138,358)(139,360)(140,359)(141,354)(142,353)(143,355)(144,356)
(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)(152,364)
(153,373)(154,374)(155,376)(156,375)(157,370)(158,369)(159,371)(160,372)
(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)(168,380)
(169,389)(170,390)(171,392)(172,391)(173,386)(174,385)(175,387)(176,388)
(177,397)(178,398)(179,400)(180,399)(181,394)(182,393)(183,395)(184,396)
(185,405)(186,406)(187,408)(188,407)(189,402)(190,401)(191,403)(192,404)
(193,413)(194,414)(195,416)(196,415)(197,410)(198,409)(199,411)(200,412)
(201,421)(202,422)(203,424)(204,423)(205,418)(206,417)(207,419)(208,420)
(209,429)(210,430)(211,432)(212,431)(213,426)(214,425)(215,427)(216,428);;
s1 := (  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)( 13, 20)
( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)( 29, 68)
( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)( 37, 60)
( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)( 45, 52)
( 46, 51)( 47, 56)( 48, 55)( 73,209)( 74,210)( 75,214)( 76,213)( 77,212)
( 78,211)( 79,216)( 80,215)( 81,201)( 82,202)( 83,206)( 84,205)( 85,204)
( 86,203)( 87,208)( 88,207)( 89,193)( 90,194)( 91,198)( 92,197)( 93,196)
( 94,195)( 95,200)( 96,199)( 97,185)( 98,186)( 99,190)(100,189)(101,188)
(102,187)(103,192)(104,191)(105,177)(106,178)(107,182)(108,181)(109,180)
(110,179)(111,184)(112,183)(113,169)(114,170)(115,174)(116,173)(117,172)
(118,171)(119,176)(120,175)(121,161)(122,162)(123,166)(124,165)(125,164)
(126,163)(127,168)(128,167)(129,153)(130,154)(131,158)(132,157)(133,156)
(134,155)(135,160)(136,159)(137,145)(138,146)(139,150)(140,149)(141,148)
(142,147)(143,152)(144,151)(217,218)(219,221)(220,222)(225,234)(226,233)
(227,237)(228,238)(229,235)(230,236)(231,239)(232,240)(241,282)(242,281)
(243,285)(244,286)(245,283)(246,284)(247,287)(248,288)(249,274)(250,273)
(251,277)(252,278)(253,275)(254,276)(255,279)(256,280)(257,266)(258,265)
(259,269)(260,270)(261,267)(262,268)(263,271)(264,272)(289,426)(290,425)
(291,429)(292,430)(293,427)(294,428)(295,431)(296,432)(297,418)(298,417)
(299,421)(300,422)(301,419)(302,420)(303,423)(304,424)(305,410)(306,409)
(307,413)(308,414)(309,411)(310,412)(311,415)(312,416)(313,402)(314,401)
(315,405)(316,406)(317,403)(318,404)(319,407)(320,408)(321,394)(322,393)
(323,397)(324,398)(325,395)(326,396)(327,399)(328,400)(329,386)(330,385)
(331,389)(332,390)(333,387)(334,388)(335,391)(336,392)(337,378)(338,377)
(339,381)(340,382)(341,379)(342,380)(343,383)(344,384)(345,370)(346,369)
(347,373)(348,374)(349,371)(350,372)(351,375)(352,376)(353,362)(354,361)
(355,365)(356,366)(357,363)(358,364)(359,367)(360,368);;
s2 := (  1,145)(  2,146)(  3,151)(  4,152)(  5,150)(  6,149)(  7,147)(  8,148)
(  9,161)( 10,162)( 11,167)( 12,168)( 13,166)( 14,165)( 15,163)( 16,164)
( 17,153)( 18,154)( 19,159)( 20,160)( 21,158)( 22,157)( 23,155)( 24,156)
( 25,209)( 26,210)( 27,215)( 28,216)( 29,214)( 30,213)( 31,211)( 32,212)
( 33,201)( 34,202)( 35,207)( 36,208)( 37,206)( 38,205)( 39,203)( 40,204)
( 41,193)( 42,194)( 43,199)( 44,200)( 45,198)( 46,197)( 47,195)( 48,196)
( 49,185)( 50,186)( 51,191)( 52,192)( 53,190)( 54,189)( 55,187)( 56,188)
( 57,177)( 58,178)( 59,183)( 60,184)( 61,182)( 62,181)( 63,179)( 64,180)
( 65,169)( 66,170)( 67,175)( 68,176)( 69,174)( 70,173)( 71,171)( 72,172)
( 75, 79)( 76, 80)( 77, 78)( 81, 89)( 82, 90)( 83, 95)( 84, 96)( 85, 94)
( 86, 93)( 87, 91)( 88, 92)( 97,137)( 98,138)( 99,143)(100,144)(101,142)
(102,141)(103,139)(104,140)(105,129)(106,130)(107,135)(108,136)(109,134)
(110,133)(111,131)(112,132)(113,121)(114,122)(115,127)(116,128)(117,126)
(118,125)(119,123)(120,124)(217,362)(218,361)(219,368)(220,367)(221,365)
(222,366)(223,364)(224,363)(225,378)(226,377)(227,384)(228,383)(229,381)
(230,382)(231,380)(232,379)(233,370)(234,369)(235,376)(236,375)(237,373)
(238,374)(239,372)(240,371)(241,426)(242,425)(243,432)(244,431)(245,429)
(246,430)(247,428)(248,427)(249,418)(250,417)(251,424)(252,423)(253,421)
(254,422)(255,420)(256,419)(257,410)(258,409)(259,416)(260,415)(261,413)
(262,414)(263,412)(264,411)(265,402)(266,401)(267,408)(268,407)(269,405)
(270,406)(271,404)(272,403)(273,394)(274,393)(275,400)(276,399)(277,397)
(278,398)(279,396)(280,395)(281,386)(282,385)(283,392)(284,391)(285,389)
(286,390)(287,388)(288,387)(289,290)(291,296)(292,295)(297,306)(298,305)
(299,312)(300,311)(301,309)(302,310)(303,308)(304,307)(313,354)(314,353)
(315,360)(316,359)(317,357)(318,358)(319,356)(320,355)(321,346)(322,345)
(323,352)(324,351)(325,349)(326,350)(327,348)(328,347)(329,338)(330,337)
(331,344)(332,343)(333,341)(334,342)(335,340)(336,339);;
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 ];;
poly := F / rels;;
 
Permutation Representation (Magma) :
s0 := Sym(432)!(  1,221)(  2,222)(  3,224)(  4,223)(  5,218)(  6,217)(  7,219)
(  8,220)(  9,229)( 10,230)( 11,232)( 12,231)( 13,226)( 14,225)( 15,227)
( 16,228)( 17,237)( 18,238)( 19,240)( 20,239)( 21,234)( 22,233)( 23,235)
( 24,236)( 25,245)( 26,246)( 27,248)( 28,247)( 29,242)( 30,241)( 31,243)
( 32,244)( 33,253)( 34,254)( 35,256)( 36,255)( 37,250)( 38,249)( 39,251)
( 40,252)( 41,261)( 42,262)( 43,264)( 44,263)( 45,258)( 46,257)( 47,259)
( 48,260)( 49,269)( 50,270)( 51,272)( 52,271)( 53,266)( 54,265)( 55,267)
( 56,268)( 57,277)( 58,278)( 59,280)( 60,279)( 61,274)( 62,273)( 63,275)
( 64,276)( 65,285)( 66,286)( 67,288)( 68,287)( 69,282)( 70,281)( 71,283)
( 72,284)( 73,293)( 74,294)( 75,296)( 76,295)( 77,290)( 78,289)( 79,291)
( 80,292)( 81,301)( 82,302)( 83,304)( 84,303)( 85,298)( 86,297)( 87,299)
( 88,300)( 89,309)( 90,310)( 91,312)( 92,311)( 93,306)( 94,305)( 95,307)
( 96,308)( 97,317)( 98,318)( 99,320)(100,319)(101,314)(102,313)(103,315)
(104,316)(105,325)(106,326)(107,328)(108,327)(109,322)(110,321)(111,323)
(112,324)(113,333)(114,334)(115,336)(116,335)(117,330)(118,329)(119,331)
(120,332)(121,341)(122,342)(123,344)(124,343)(125,338)(126,337)(127,339)
(128,340)(129,349)(130,350)(131,352)(132,351)(133,346)(134,345)(135,347)
(136,348)(137,357)(138,358)(139,360)(140,359)(141,354)(142,353)(143,355)
(144,356)(145,365)(146,366)(147,368)(148,367)(149,362)(150,361)(151,363)
(152,364)(153,373)(154,374)(155,376)(156,375)(157,370)(158,369)(159,371)
(160,372)(161,381)(162,382)(163,384)(164,383)(165,378)(166,377)(167,379)
(168,380)(169,389)(170,390)(171,392)(172,391)(173,386)(174,385)(175,387)
(176,388)(177,397)(178,398)(179,400)(180,399)(181,394)(182,393)(183,395)
(184,396)(185,405)(186,406)(187,408)(188,407)(189,402)(190,401)(191,403)
(192,404)(193,413)(194,414)(195,416)(196,415)(197,410)(198,409)(199,411)
(200,412)(201,421)(202,422)(203,424)(204,423)(205,418)(206,417)(207,419)
(208,420)(209,429)(210,430)(211,432)(212,431)(213,426)(214,425)(215,427)
(216,428);
s1 := Sym(432)!(  3,  6)(  4,  5)(  7,  8)(  9, 17)( 10, 18)( 11, 22)( 12, 21)
( 13, 20)( 14, 19)( 15, 24)( 16, 23)( 25, 65)( 26, 66)( 27, 70)( 28, 69)
( 29, 68)( 30, 67)( 31, 72)( 32, 71)( 33, 57)( 34, 58)( 35, 62)( 36, 61)
( 37, 60)( 38, 59)( 39, 64)( 40, 63)( 41, 49)( 42, 50)( 43, 54)( 44, 53)
( 45, 52)( 46, 51)( 47, 56)( 48, 55)( 73,209)( 74,210)( 75,214)( 76,213)
( 77,212)( 78,211)( 79,216)( 80,215)( 81,201)( 82,202)( 83,206)( 84,205)
( 85,204)( 86,203)( 87,208)( 88,207)( 89,193)( 90,194)( 91,198)( 92,197)
( 93,196)( 94,195)( 95,200)( 96,199)( 97,185)( 98,186)( 99,190)(100,189)
(101,188)(102,187)(103,192)(104,191)(105,177)(106,178)(107,182)(108,181)
(109,180)(110,179)(111,184)(112,183)(113,169)(114,170)(115,174)(116,173)
(117,172)(118,171)(119,176)(120,175)(121,161)(122,162)(123,166)(124,165)
(125,164)(126,163)(127,168)(128,167)(129,153)(130,154)(131,158)(132,157)
(133,156)(134,155)(135,160)(136,159)(137,145)(138,146)(139,150)(140,149)
(141,148)(142,147)(143,152)(144,151)(217,218)(219,221)(220,222)(225,234)
(226,233)(227,237)(228,238)(229,235)(230,236)(231,239)(232,240)(241,282)
(242,281)(243,285)(244,286)(245,283)(246,284)(247,287)(248,288)(249,274)
(250,273)(251,277)(252,278)(253,275)(254,276)(255,279)(256,280)(257,266)
(258,265)(259,269)(260,270)(261,267)(262,268)(263,271)(264,272)(289,426)
(290,425)(291,429)(292,430)(293,427)(294,428)(295,431)(296,432)(297,418)
(298,417)(299,421)(300,422)(301,419)(302,420)(303,423)(304,424)(305,410)
(306,409)(307,413)(308,414)(309,411)(310,412)(311,415)(312,416)(313,402)
(314,401)(315,405)(316,406)(317,403)(318,404)(319,407)(320,408)(321,394)
(322,393)(323,397)(324,398)(325,395)(326,396)(327,399)(328,400)(329,386)
(330,385)(331,389)(332,390)(333,387)(334,388)(335,391)(336,392)(337,378)
(338,377)(339,381)(340,382)(341,379)(342,380)(343,383)(344,384)(345,370)
(346,369)(347,373)(348,374)(349,371)(350,372)(351,375)(352,376)(353,362)
(354,361)(355,365)(356,366)(357,363)(358,364)(359,367)(360,368);
s2 := Sym(432)!(  1,145)(  2,146)(  3,151)(  4,152)(  5,150)(  6,149)(  7,147)
(  8,148)(  9,161)( 10,162)( 11,167)( 12,168)( 13,166)( 14,165)( 15,163)
( 16,164)( 17,153)( 18,154)( 19,159)( 20,160)( 21,158)( 22,157)( 23,155)
( 24,156)( 25,209)( 26,210)( 27,215)( 28,216)( 29,214)( 30,213)( 31,211)
( 32,212)( 33,201)( 34,202)( 35,207)( 36,208)( 37,206)( 38,205)( 39,203)
( 40,204)( 41,193)( 42,194)( 43,199)( 44,200)( 45,198)( 46,197)( 47,195)
( 48,196)( 49,185)( 50,186)( 51,191)( 52,192)( 53,190)( 54,189)( 55,187)
( 56,188)( 57,177)( 58,178)( 59,183)( 60,184)( 61,182)( 62,181)( 63,179)
( 64,180)( 65,169)( 66,170)( 67,175)( 68,176)( 69,174)( 70,173)( 71,171)
( 72,172)( 75, 79)( 76, 80)( 77, 78)( 81, 89)( 82, 90)( 83, 95)( 84, 96)
( 85, 94)( 86, 93)( 87, 91)( 88, 92)( 97,137)( 98,138)( 99,143)(100,144)
(101,142)(102,141)(103,139)(104,140)(105,129)(106,130)(107,135)(108,136)
(109,134)(110,133)(111,131)(112,132)(113,121)(114,122)(115,127)(116,128)
(117,126)(118,125)(119,123)(120,124)(217,362)(218,361)(219,368)(220,367)
(221,365)(222,366)(223,364)(224,363)(225,378)(226,377)(227,384)(228,383)
(229,381)(230,382)(231,380)(232,379)(233,370)(234,369)(235,376)(236,375)
(237,373)(238,374)(239,372)(240,371)(241,426)(242,425)(243,432)(244,431)
(245,429)(246,430)(247,428)(248,427)(249,418)(250,417)(251,424)(252,423)
(253,421)(254,422)(255,420)(256,419)(257,410)(258,409)(259,416)(260,415)
(261,413)(262,414)(263,412)(264,411)(265,402)(266,401)(267,408)(268,407)
(269,405)(270,406)(271,404)(272,403)(273,394)(274,393)(275,400)(276,399)
(277,397)(278,398)(279,396)(280,395)(281,386)(282,385)(283,392)(284,391)
(285,389)(286,390)(287,388)(288,387)(289,290)(291,296)(292,295)(297,306)
(298,305)(299,312)(300,311)(301,309)(302,310)(303,308)(304,307)(313,354)
(314,353)(315,360)(316,359)(317,357)(318,358)(319,356)(320,355)(321,346)
(322,345)(323,352)(324,351)(325,349)(326,350)(327,348)(328,347)(329,338)
(330,337)(331,344)(332,343)(333,341)(334,342)(335,340)(336,339);
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*s2*s1*s2*s1*s0*s1*s2*s1*s2*s1, 
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, 
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2 >; 
 
References : None.
to this polytope