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