Polytope of Type {54,16}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,16}*1728
Also Known As : {54,16|2}. if this polytope has another name.
Group : SmallGroup(1728,478)
Rank : 3
Schlafli Type : {54,16}
Number of vertices, edges, etc : 54, 432, 16
Order of s0s1s2 : 432
Order of s0s1s2s1 : 2
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Flat
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 : {54,8}*864
3-fold quotients : {18,16}*576
4-fold quotients : {54,4}*432a
6-fold quotients : {18,8}*288
8-fold quotients : {54,2}*216
9-fold quotients : {6,16}*192
12-fold quotients : {18,4}*144a
16-fold quotients : {27,2}*108
18-fold quotients : {6,8}*96
24-fold quotients : {18,2}*72
27-fold quotients : {2,16}*64
36-fold quotients : {6,4}*48a
48-fold quotients : {9,2}*36
54-fold quotients : {2,8}*32
72-fold quotients : {6,2}*24
108-fold quotients : {2,4}*16
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 := ( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)( 13, 24)
( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)( 32, 35)
( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)( 43, 48)
( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)( 65, 80)
( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)( 83, 84)
( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)( 95,104)
( 96,103)( 97,102)( 98,101)( 99,100)(110,111)(112,117)(113,116)(114,115)
(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)(125,128)
(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)(147,160)
(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(164,165)(166,171)
(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)(176,185)(177,184)
(178,183)(179,182)(180,181)(191,192)(193,198)(194,197)(195,196)(199,216)
(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)(207,208)
(218,219)(220,225)(221,224)(222,223)(226,243)(227,242)(228,241)(229,240)
(230,239)(231,238)(232,237)(233,236)(234,235)(245,246)(247,252)(248,251)
(249,250)(253,270)(254,269)(255,268)(256,267)(257,266)(258,265)(259,264)
(260,263)(261,262)(272,273)(274,279)(275,278)(276,277)(280,297)(281,296)
(282,295)(283,294)(284,293)(285,292)(286,291)(287,290)(288,289)(299,300)
(301,306)(302,305)(303,304)(307,324)(308,323)(309,322)(310,321)(311,320)
(312,319)(313,318)(314,317)(315,316)(326,327)(328,333)(329,332)(330,331)
(334,351)(335,350)(336,349)(337,348)(338,347)(339,346)(340,345)(341,344)
(342,343)(353,354)(355,360)(356,359)(357,358)(361,378)(362,377)(363,376)
(364,375)(365,374)(366,373)(367,372)(368,371)(369,370)(380,381)(382,387)
(383,386)(384,385)(388,405)(389,404)(390,403)(391,402)(392,401)(393,400)
(394,399)(395,398)(396,397)(407,408)(409,414)(410,413)(411,412)(415,432)
(416,431)(417,430)(418,429)(419,428)(420,427)(421,426)(422,425)(423,424);;
s1 := ( 1, 10)( 2, 12)( 3, 11)( 4, 18)( 5, 17)( 6, 16)( 7, 15)( 8, 14)
( 9, 13)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 28, 37)( 29, 39)( 30, 38)
( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 46, 54)( 47, 53)
( 48, 52)( 49, 51)( 55, 91)( 56, 93)( 57, 92)( 58, 99)( 59, 98)( 60, 97)
( 61, 96)( 62, 95)( 63, 94)( 64, 82)( 65, 84)( 66, 83)( 67, 90)( 68, 89)
( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73,108)( 74,107)( 75,106)( 76,105)
( 77,104)( 78,103)( 79,102)( 80,101)( 81,100)(109,172)(110,174)(111,173)
(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,163)(119,165)
(120,164)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)(127,189)
(128,188)(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)(135,181)
(136,199)(137,201)(138,200)(139,207)(140,206)(141,205)(142,204)(143,203)
(144,202)(145,190)(146,192)(147,191)(148,198)(149,197)(150,196)(151,195)
(152,194)(153,193)(154,216)(155,215)(156,214)(157,213)(158,212)(159,211)
(160,210)(161,209)(162,208)(217,334)(218,336)(219,335)(220,342)(221,341)
(222,340)(223,339)(224,338)(225,337)(226,325)(227,327)(228,326)(229,333)
(230,332)(231,331)(232,330)(233,329)(234,328)(235,351)(236,350)(237,349)
(238,348)(239,347)(240,346)(241,345)(242,344)(243,343)(244,361)(245,363)
(246,362)(247,369)(248,368)(249,367)(250,366)(251,365)(252,364)(253,352)
(254,354)(255,353)(256,360)(257,359)(258,358)(259,357)(260,356)(261,355)
(262,378)(263,377)(264,376)(265,375)(266,374)(267,373)(268,372)(269,371)
(270,370)(271,415)(272,417)(273,416)(274,423)(275,422)(276,421)(277,420)
(278,419)(279,418)(280,406)(281,408)(282,407)(283,414)(284,413)(285,412)
(286,411)(287,410)(288,409)(289,432)(290,431)(291,430)(292,429)(293,428)
(294,427)(295,426)(296,425)(297,424)(298,388)(299,390)(300,389)(301,396)
(302,395)(303,394)(304,393)(305,392)(306,391)(307,379)(308,381)(309,380)
(310,387)(311,386)(312,385)(313,384)(314,383)(315,382)(316,405)(317,404)
(318,403)(319,402)(320,401)(321,400)(322,399)(323,398)(324,397);;
s2 := ( 1,217)( 2,218)( 3,219)( 4,220)( 5,221)( 6,222)( 7,223)( 8,224)
( 9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)( 16,232)
( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)( 24,240)
( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)( 32,248)
( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)( 40,256)
( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)( 48,264)
( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,298)( 56,299)
( 57,300)( 58,301)( 59,302)( 60,303)( 61,304)( 62,305)( 63,306)( 64,307)
( 65,308)( 66,309)( 67,310)( 68,311)( 69,312)( 70,313)( 71,314)( 72,315)
( 73,316)( 74,317)( 75,318)( 76,319)( 77,320)( 78,321)( 79,322)( 80,323)
( 81,324)( 82,271)( 83,272)( 84,273)( 85,274)( 86,275)( 87,276)( 88,277)
( 89,278)( 90,279)( 91,280)( 92,281)( 93,282)( 94,283)( 95,284)( 96,285)
( 97,286)( 98,287)( 99,288)(100,289)(101,290)(102,291)(103,292)(104,293)
(105,294)(106,295)(107,296)(108,297)(109,379)(110,380)(111,381)(112,382)
(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)(120,390)
(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)(128,398)
(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)(136,406)
(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)(144,414)
(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)(152,422)
(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)(160,430)
(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)(168,330)
(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)(176,338)
(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)(184,346)
(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)(192,354)
(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)(200,362)
(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)(208,370)
(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)(216,378);;
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*s2*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,
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)!( 2, 3)( 4, 9)( 5, 8)( 6, 7)( 10, 27)( 11, 26)( 12, 25)
( 13, 24)( 14, 23)( 15, 22)( 16, 21)( 17, 20)( 18, 19)( 29, 30)( 31, 36)
( 32, 35)( 33, 34)( 37, 54)( 38, 53)( 39, 52)( 40, 51)( 41, 50)( 42, 49)
( 43, 48)( 44, 47)( 45, 46)( 56, 57)( 58, 63)( 59, 62)( 60, 61)( 64, 81)
( 65, 80)( 66, 79)( 67, 78)( 68, 77)( 69, 76)( 70, 75)( 71, 74)( 72, 73)
( 83, 84)( 85, 90)( 86, 89)( 87, 88)( 91,108)( 92,107)( 93,106)( 94,105)
( 95,104)( 96,103)( 97,102)( 98,101)( 99,100)(110,111)(112,117)(113,116)
(114,115)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)
(125,128)(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)
(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(164,165)
(166,171)(167,170)(168,169)(172,189)(173,188)(174,187)(175,186)(176,185)
(177,184)(178,183)(179,182)(180,181)(191,192)(193,198)(194,197)(195,196)
(199,216)(200,215)(201,214)(202,213)(203,212)(204,211)(205,210)(206,209)
(207,208)(218,219)(220,225)(221,224)(222,223)(226,243)(227,242)(228,241)
(229,240)(230,239)(231,238)(232,237)(233,236)(234,235)(245,246)(247,252)
(248,251)(249,250)(253,270)(254,269)(255,268)(256,267)(257,266)(258,265)
(259,264)(260,263)(261,262)(272,273)(274,279)(275,278)(276,277)(280,297)
(281,296)(282,295)(283,294)(284,293)(285,292)(286,291)(287,290)(288,289)
(299,300)(301,306)(302,305)(303,304)(307,324)(308,323)(309,322)(310,321)
(311,320)(312,319)(313,318)(314,317)(315,316)(326,327)(328,333)(329,332)
(330,331)(334,351)(335,350)(336,349)(337,348)(338,347)(339,346)(340,345)
(341,344)(342,343)(353,354)(355,360)(356,359)(357,358)(361,378)(362,377)
(363,376)(364,375)(365,374)(366,373)(367,372)(368,371)(369,370)(380,381)
(382,387)(383,386)(384,385)(388,405)(389,404)(390,403)(391,402)(392,401)
(393,400)(394,399)(395,398)(396,397)(407,408)(409,414)(410,413)(411,412)
(415,432)(416,431)(417,430)(418,429)(419,428)(420,427)(421,426)(422,425)
(423,424);
s1 := Sym(432)!( 1, 10)( 2, 12)( 3, 11)( 4, 18)( 5, 17)( 6, 16)( 7, 15)
( 8, 14)( 9, 13)( 19, 27)( 20, 26)( 21, 25)( 22, 24)( 28, 37)( 29, 39)
( 30, 38)( 31, 45)( 32, 44)( 33, 43)( 34, 42)( 35, 41)( 36, 40)( 46, 54)
( 47, 53)( 48, 52)( 49, 51)( 55, 91)( 56, 93)( 57, 92)( 58, 99)( 59, 98)
( 60, 97)( 61, 96)( 62, 95)( 63, 94)( 64, 82)( 65, 84)( 66, 83)( 67, 90)
( 68, 89)( 69, 88)( 70, 87)( 71, 86)( 72, 85)( 73,108)( 74,107)( 75,106)
( 76,105)( 77,104)( 78,103)( 79,102)( 80,101)( 81,100)(109,172)(110,174)
(111,173)(112,180)(113,179)(114,178)(115,177)(116,176)(117,175)(118,163)
(119,165)(120,164)(121,171)(122,170)(123,169)(124,168)(125,167)(126,166)
(127,189)(128,188)(129,187)(130,186)(131,185)(132,184)(133,183)(134,182)
(135,181)(136,199)(137,201)(138,200)(139,207)(140,206)(141,205)(142,204)
(143,203)(144,202)(145,190)(146,192)(147,191)(148,198)(149,197)(150,196)
(151,195)(152,194)(153,193)(154,216)(155,215)(156,214)(157,213)(158,212)
(159,211)(160,210)(161,209)(162,208)(217,334)(218,336)(219,335)(220,342)
(221,341)(222,340)(223,339)(224,338)(225,337)(226,325)(227,327)(228,326)
(229,333)(230,332)(231,331)(232,330)(233,329)(234,328)(235,351)(236,350)
(237,349)(238,348)(239,347)(240,346)(241,345)(242,344)(243,343)(244,361)
(245,363)(246,362)(247,369)(248,368)(249,367)(250,366)(251,365)(252,364)
(253,352)(254,354)(255,353)(256,360)(257,359)(258,358)(259,357)(260,356)
(261,355)(262,378)(263,377)(264,376)(265,375)(266,374)(267,373)(268,372)
(269,371)(270,370)(271,415)(272,417)(273,416)(274,423)(275,422)(276,421)
(277,420)(278,419)(279,418)(280,406)(281,408)(282,407)(283,414)(284,413)
(285,412)(286,411)(287,410)(288,409)(289,432)(290,431)(291,430)(292,429)
(293,428)(294,427)(295,426)(296,425)(297,424)(298,388)(299,390)(300,389)
(301,396)(302,395)(303,394)(304,393)(305,392)(306,391)(307,379)(308,381)
(309,380)(310,387)(311,386)(312,385)(313,384)(314,383)(315,382)(316,405)
(317,404)(318,403)(319,402)(320,401)(321,400)(322,399)(323,398)(324,397);
s2 := Sym(432)!( 1,217)( 2,218)( 3,219)( 4,220)( 5,221)( 6,222)( 7,223)
( 8,224)( 9,225)( 10,226)( 11,227)( 12,228)( 13,229)( 14,230)( 15,231)
( 16,232)( 17,233)( 18,234)( 19,235)( 20,236)( 21,237)( 22,238)( 23,239)
( 24,240)( 25,241)( 26,242)( 27,243)( 28,244)( 29,245)( 30,246)( 31,247)
( 32,248)( 33,249)( 34,250)( 35,251)( 36,252)( 37,253)( 38,254)( 39,255)
( 40,256)( 41,257)( 42,258)( 43,259)( 44,260)( 45,261)( 46,262)( 47,263)
( 48,264)( 49,265)( 50,266)( 51,267)( 52,268)( 53,269)( 54,270)( 55,298)
( 56,299)( 57,300)( 58,301)( 59,302)( 60,303)( 61,304)( 62,305)( 63,306)
( 64,307)( 65,308)( 66,309)( 67,310)( 68,311)( 69,312)( 70,313)( 71,314)
( 72,315)( 73,316)( 74,317)( 75,318)( 76,319)( 77,320)( 78,321)( 79,322)
( 80,323)( 81,324)( 82,271)( 83,272)( 84,273)( 85,274)( 86,275)( 87,276)
( 88,277)( 89,278)( 90,279)( 91,280)( 92,281)( 93,282)( 94,283)( 95,284)
( 96,285)( 97,286)( 98,287)( 99,288)(100,289)(101,290)(102,291)(103,292)
(104,293)(105,294)(106,295)(107,296)(108,297)(109,379)(110,380)(111,381)
(112,382)(113,383)(114,384)(115,385)(116,386)(117,387)(118,388)(119,389)
(120,390)(121,391)(122,392)(123,393)(124,394)(125,395)(126,396)(127,397)
(128,398)(129,399)(130,400)(131,401)(132,402)(133,403)(134,404)(135,405)
(136,406)(137,407)(138,408)(139,409)(140,410)(141,411)(142,412)(143,413)
(144,414)(145,415)(146,416)(147,417)(148,418)(149,419)(150,420)(151,421)
(152,422)(153,423)(154,424)(155,425)(156,426)(157,427)(158,428)(159,429)
(160,430)(161,431)(162,432)(163,325)(164,326)(165,327)(166,328)(167,329)
(168,330)(169,331)(170,332)(171,333)(172,334)(173,335)(174,336)(175,337)
(176,338)(177,339)(178,340)(179,341)(180,342)(181,343)(182,344)(183,345)
(184,346)(185,347)(186,348)(187,349)(188,350)(189,351)(190,352)(191,353)
(192,354)(193,355)(194,356)(195,357)(196,358)(197,359)(198,360)(199,361)
(200,362)(201,363)(202,364)(203,365)(204,366)(205,367)(206,368)(207,369)
(208,370)(209,371)(210,372)(211,373)(212,374)(213,375)(214,376)(215,377)
(216,378);
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*s2*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,
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