Polytope of Type {8,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,54}*1728c
if this polytope has a name.
Group : SmallGroup(1728,11371)
Rank : 3
Schlafli Type : {8,54}
Number of vertices, edges, etc : 16, 432, 108
Order of s0s1s2 : 54
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {4,54}*864
3-fold quotients : {8,18}*576c
4-fold quotients : {4,27}*432, {4,54}*432b, {4,54}*432c
6-fold quotients : {4,18}*288
8-fold quotients : {4,27}*216, {2,54}*216
9-fold quotients : {8,6}*192c
12-fold quotients : {4,9}*144, {4,18}*144b, {4,18}*144c
16-fold quotients : {2,27}*108
18-fold quotients : {4,6}*96
24-fold quotients : {4,9}*72, {2,18}*72
36-fold quotients : {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {2,9}*36
72-fold quotients : {4,3}*24, {2,6}*24
144-fold quotients : {2,3}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
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, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)( 13, 23)
( 14, 24)( 15, 21)( 16, 22)( 25, 65)( 26, 66)( 27, 68)( 28, 67)( 29, 71)
( 30, 72)( 31, 69)( 32, 70)( 33, 57)( 34, 58)( 35, 60)( 36, 59)( 37, 63)
( 38, 64)( 39, 61)( 40, 62)( 41, 49)( 42, 50)( 43, 52)( 44, 51)( 45, 55)
( 46, 56)( 47, 53)( 48, 54)( 73,209)( 74,210)( 75,212)( 76,211)( 77,215)
( 78,216)( 79,213)( 80,214)( 81,201)( 82,202)( 83,204)( 84,203)( 85,207)
( 86,208)( 87,205)( 88,206)( 89,193)( 90,194)( 91,196)( 92,195)( 93,199)
( 94,200)( 95,197)( 96,198)( 97,185)( 98,186)( 99,188)(100,187)(101,191)
(102,192)(103,189)(104,190)(105,177)(106,178)(107,180)(108,179)(109,183)
(110,184)(111,181)(112,182)(113,169)(114,170)(115,172)(116,171)(117,175)
(118,176)(119,173)(120,174)(121,161)(122,162)(123,164)(124,163)(125,167)
(126,168)(127,165)(128,166)(129,153)(130,154)(131,156)(132,155)(133,159)
(134,160)(135,157)(136,158)(137,145)(138,146)(139,148)(140,147)(141,151)
(142,152)(143,149)(144,150)(219,220)(221,223)(222,224)(225,233)(226,234)
(227,236)(228,235)(229,239)(230,240)(231,237)(232,238)(241,281)(242,282)
(243,284)(244,283)(245,287)(246,288)(247,285)(248,286)(249,273)(250,274)
(251,276)(252,275)(253,279)(254,280)(255,277)(256,278)(257,265)(258,266)
(259,268)(260,267)(261,271)(262,272)(263,269)(264,270)(289,425)(290,426)
(291,428)(292,427)(293,431)(294,432)(295,429)(296,430)(297,417)(298,418)
(299,420)(300,419)(301,423)(302,424)(303,421)(304,422)(305,409)(306,410)
(307,412)(308,411)(309,415)(310,416)(311,413)(312,414)(313,401)(314,402)
(315,404)(316,403)(317,407)(318,408)(319,405)(320,406)(321,393)(322,394)
(323,396)(324,395)(325,399)(326,400)(327,397)(328,398)(329,385)(330,386)
(331,388)(332,387)(333,391)(334,392)(335,389)(336,390)(337,377)(338,378)
(339,380)(340,379)(341,383)(342,384)(343,381)(344,382)(345,369)(346,370)
(347,372)(348,371)(349,375)(350,376)(351,373)(352,374)(353,361)(354,362)
(355,364)(356,363)(357,367)(358,368)(359,365)(360,366);;
s2 := ( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)( 8, 76)
( 9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)( 16, 92)
( 17, 81)( 18, 82)( 19, 87)( 20, 88)( 21, 86)( 22, 85)( 23, 83)( 24, 84)
( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)( 32,140)
( 33,129)( 34,130)( 35,135)( 36,136)( 37,134)( 38,133)( 39,131)( 40,132)
( 41,121)( 42,122)( 43,127)( 44,128)( 45,126)( 46,125)( 47,123)( 48,124)
( 49,113)( 50,114)( 51,119)( 52,120)( 53,118)( 54,117)( 55,115)( 56,116)
( 57,105)( 58,106)( 59,111)( 60,112)( 61,110)( 62,109)( 63,107)( 64,108)
( 65, 97)( 66, 98)( 67,103)( 68,104)( 69,102)( 70,101)( 71, 99)( 72,100)
(145,209)(146,210)(147,215)(148,216)(149,214)(150,213)(151,211)(152,212)
(153,201)(154,202)(155,207)(156,208)(157,206)(158,205)(159,203)(160,204)
(161,193)(162,194)(163,199)(164,200)(165,198)(166,197)(167,195)(168,196)
(169,185)(170,186)(171,191)(172,192)(173,190)(174,189)(175,187)(176,188)
(179,183)(180,184)(181,182)(217,290)(218,289)(219,296)(220,295)(221,293)
(222,294)(223,292)(224,291)(225,306)(226,305)(227,312)(228,311)(229,309)
(230,310)(231,308)(232,307)(233,298)(234,297)(235,304)(236,303)(237,301)
(238,302)(239,300)(240,299)(241,354)(242,353)(243,360)(244,359)(245,357)
(246,358)(247,356)(248,355)(249,346)(250,345)(251,352)(252,351)(253,349)
(254,350)(255,348)(256,347)(257,338)(258,337)(259,344)(260,343)(261,341)
(262,342)(263,340)(264,339)(265,330)(266,329)(267,336)(268,335)(269,333)
(270,334)(271,332)(272,331)(273,322)(274,321)(275,328)(276,327)(277,325)
(278,326)(279,324)(280,323)(281,314)(282,313)(283,320)(284,319)(285,317)
(286,318)(287,316)(288,315)(361,426)(362,425)(363,432)(364,431)(365,429)
(366,430)(367,428)(368,427)(369,418)(370,417)(371,424)(372,423)(373,421)
(374,422)(375,420)(376,419)(377,410)(378,409)(379,416)(380,415)(381,413)
(382,414)(383,412)(384,411)(385,402)(386,401)(387,408)(388,407)(389,405)
(390,406)(391,404)(392,403)(393,394)(395,400)(396,399);;
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, s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*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*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, 4)( 5, 7)( 6, 8)( 9, 17)( 10, 18)( 11, 20)( 12, 19)
( 13, 23)( 14, 24)( 15, 21)( 16, 22)( 25, 65)( 26, 66)( 27, 68)( 28, 67)
( 29, 71)( 30, 72)( 31, 69)( 32, 70)( 33, 57)( 34, 58)( 35, 60)( 36, 59)
( 37, 63)( 38, 64)( 39, 61)( 40, 62)( 41, 49)( 42, 50)( 43, 52)( 44, 51)
( 45, 55)( 46, 56)( 47, 53)( 48, 54)( 73,209)( 74,210)( 75,212)( 76,211)
( 77,215)( 78,216)( 79,213)( 80,214)( 81,201)( 82,202)( 83,204)( 84,203)
( 85,207)( 86,208)( 87,205)( 88,206)( 89,193)( 90,194)( 91,196)( 92,195)
( 93,199)( 94,200)( 95,197)( 96,198)( 97,185)( 98,186)( 99,188)(100,187)
(101,191)(102,192)(103,189)(104,190)(105,177)(106,178)(107,180)(108,179)
(109,183)(110,184)(111,181)(112,182)(113,169)(114,170)(115,172)(116,171)
(117,175)(118,176)(119,173)(120,174)(121,161)(122,162)(123,164)(124,163)
(125,167)(126,168)(127,165)(128,166)(129,153)(130,154)(131,156)(132,155)
(133,159)(134,160)(135,157)(136,158)(137,145)(138,146)(139,148)(140,147)
(141,151)(142,152)(143,149)(144,150)(219,220)(221,223)(222,224)(225,233)
(226,234)(227,236)(228,235)(229,239)(230,240)(231,237)(232,238)(241,281)
(242,282)(243,284)(244,283)(245,287)(246,288)(247,285)(248,286)(249,273)
(250,274)(251,276)(252,275)(253,279)(254,280)(255,277)(256,278)(257,265)
(258,266)(259,268)(260,267)(261,271)(262,272)(263,269)(264,270)(289,425)
(290,426)(291,428)(292,427)(293,431)(294,432)(295,429)(296,430)(297,417)
(298,418)(299,420)(300,419)(301,423)(302,424)(303,421)(304,422)(305,409)
(306,410)(307,412)(308,411)(309,415)(310,416)(311,413)(312,414)(313,401)
(314,402)(315,404)(316,403)(317,407)(318,408)(319,405)(320,406)(321,393)
(322,394)(323,396)(324,395)(325,399)(326,400)(327,397)(328,398)(329,385)
(330,386)(331,388)(332,387)(333,391)(334,392)(335,389)(336,390)(337,377)
(338,378)(339,380)(340,379)(341,383)(342,384)(343,381)(344,382)(345,369)
(346,370)(347,372)(348,371)(349,375)(350,376)(351,373)(352,374)(353,361)
(354,362)(355,364)(356,363)(357,367)(358,368)(359,365)(360,366);
s2 := Sym(432)!( 1, 73)( 2, 74)( 3, 79)( 4, 80)( 5, 78)( 6, 77)( 7, 75)
( 8, 76)( 9, 89)( 10, 90)( 11, 95)( 12, 96)( 13, 94)( 14, 93)( 15, 91)
( 16, 92)( 17, 81)( 18, 82)( 19, 87)( 20, 88)( 21, 86)( 22, 85)( 23, 83)
( 24, 84)( 25,137)( 26,138)( 27,143)( 28,144)( 29,142)( 30,141)( 31,139)
( 32,140)( 33,129)( 34,130)( 35,135)( 36,136)( 37,134)( 38,133)( 39,131)
( 40,132)( 41,121)( 42,122)( 43,127)( 44,128)( 45,126)( 46,125)( 47,123)
( 48,124)( 49,113)( 50,114)( 51,119)( 52,120)( 53,118)( 54,117)( 55,115)
( 56,116)( 57,105)( 58,106)( 59,111)( 60,112)( 61,110)( 62,109)( 63,107)
( 64,108)( 65, 97)( 66, 98)( 67,103)( 68,104)( 69,102)( 70,101)( 71, 99)
( 72,100)(145,209)(146,210)(147,215)(148,216)(149,214)(150,213)(151,211)
(152,212)(153,201)(154,202)(155,207)(156,208)(157,206)(158,205)(159,203)
(160,204)(161,193)(162,194)(163,199)(164,200)(165,198)(166,197)(167,195)
(168,196)(169,185)(170,186)(171,191)(172,192)(173,190)(174,189)(175,187)
(176,188)(179,183)(180,184)(181,182)(217,290)(218,289)(219,296)(220,295)
(221,293)(222,294)(223,292)(224,291)(225,306)(226,305)(227,312)(228,311)
(229,309)(230,310)(231,308)(232,307)(233,298)(234,297)(235,304)(236,303)
(237,301)(238,302)(239,300)(240,299)(241,354)(242,353)(243,360)(244,359)
(245,357)(246,358)(247,356)(248,355)(249,346)(250,345)(251,352)(252,351)
(253,349)(254,350)(255,348)(256,347)(257,338)(258,337)(259,344)(260,343)
(261,341)(262,342)(263,340)(264,339)(265,330)(266,329)(267,336)(268,335)
(269,333)(270,334)(271,332)(272,331)(273,322)(274,321)(275,328)(276,327)
(277,325)(278,326)(279,324)(280,323)(281,314)(282,313)(283,320)(284,319)
(285,317)(286,318)(287,316)(288,315)(361,426)(362,425)(363,432)(364,431)
(365,429)(366,430)(367,428)(368,427)(369,418)(370,417)(371,424)(372,423)
(373,421)(374,422)(375,420)(376,419)(377,410)(378,409)(379,416)(380,415)
(381,413)(382,414)(383,412)(384,411)(385,402)(386,401)(387,408)(388,407)
(389,405)(390,406)(391,404)(392,403)(393,394)(395,400)(396,399);
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, s2*s0*s1*s2*s1*s0*s1*s2*s1*s2*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*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