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