Polytope of Type {16,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {16,54}*1728
Also Known As : {16,54|2}. if this polytope has another name.
Group : SmallGroup(1728,478)
Rank : 3
Schlafli Type : {16,54}
Number of vertices, edges, etc : 16, 432, 54
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 : {8,54}*864
3-fold quotients : {16,18}*576
4-fold quotients : {4,54}*432a
6-fold quotients : {8,18}*288
8-fold quotients : {2,54}*216
9-fold quotients : {16,6}*192
12-fold quotients : {4,18}*144a
16-fold quotients : {2,27}*108
18-fold quotients : {8,6}*96
24-fold quotients : {2,18}*72
27-fold quotients : {16,2}*64
36-fold quotients : {4,6}*48a
48-fold quotients : {2,9}*36
54-fold quotients : {8,2}*32
72-fold quotients : {2,6}*24
108-fold quotients : {4,2}*16
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 := ( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)( 62, 89)
( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)( 70, 97)
( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)( 78,105)
( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)(113,167)
(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)(121,175)
(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)(129,183)
(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)(137,191)
(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)(145,199)
(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)(153,207)
(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)(161,215)
(162,216)(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)
(224,332)(225,333)(226,334)(227,335)(228,336)(229,337)(230,338)(231,339)
(232,340)(233,341)(234,342)(235,343)(236,344)(237,345)(238,346)(239,347)
(240,348)(241,349)(242,350)(243,351)(244,352)(245,353)(246,354)(247,355)
(248,356)(249,357)(250,358)(251,359)(252,360)(253,361)(254,362)(255,363)
(256,364)(257,365)(258,366)(259,367)(260,368)(261,369)(262,370)(263,371)
(264,372)(265,373)(266,374)(267,375)(268,376)(269,377)(270,378)(271,406)
(272,407)(273,408)(274,409)(275,410)(276,411)(277,412)(278,413)(279,414)
(280,415)(281,416)(282,417)(283,418)(284,419)(285,420)(286,421)(287,422)
(288,423)(289,424)(290,425)(291,426)(292,427)(293,428)(294,429)(295,430)
(296,431)(297,432)(298,379)(299,380)(300,381)(301,382)(302,383)(303,384)
(304,385)(305,386)(306,387)(307,388)(308,389)(309,390)(310,391)(311,392)
(312,393)(313,394)(314,395)(315,396)(316,397)(317,398)(318,399)(319,400)
(320,401)(321,402)(322,403)(323,404)(324,405);;
s1 := ( 1,217)( 2,219)( 3,218)( 4,225)( 5,224)( 6,223)( 7,222)( 8,221)
( 9,220)( 10,243)( 11,242)( 12,241)( 13,240)( 14,239)( 15,238)( 16,237)
( 17,236)( 18,235)( 19,234)( 20,233)( 21,232)( 22,231)( 23,230)( 24,229)
( 25,228)( 26,227)( 27,226)( 28,244)( 29,246)( 30,245)( 31,252)( 32,251)
( 33,250)( 34,249)( 35,248)( 36,247)( 37,270)( 38,269)( 39,268)( 40,267)
( 41,266)( 42,265)( 43,264)( 44,263)( 45,262)( 46,261)( 47,260)( 48,259)
( 49,258)( 50,257)( 51,256)( 52,255)( 53,254)( 54,253)( 55,298)( 56,300)
( 57,299)( 58,306)( 59,305)( 60,304)( 61,303)( 62,302)( 63,301)( 64,324)
( 65,323)( 66,322)( 67,321)( 68,320)( 69,319)( 70,318)( 71,317)( 72,316)
( 73,315)( 74,314)( 75,313)( 76,312)( 77,311)( 78,310)( 79,309)( 80,308)
( 81,307)( 82,271)( 83,273)( 84,272)( 85,279)( 86,278)( 87,277)( 88,276)
( 89,275)( 90,274)( 91,297)( 92,296)( 93,295)( 94,294)( 95,293)( 96,292)
( 97,291)( 98,290)( 99,289)(100,288)(101,287)(102,286)(103,285)(104,284)
(105,283)(106,282)(107,281)(108,280)(109,379)(110,381)(111,380)(112,387)
(113,386)(114,385)(115,384)(116,383)(117,382)(118,405)(119,404)(120,403)
(121,402)(122,401)(123,400)(124,399)(125,398)(126,397)(127,396)(128,395)
(129,394)(130,393)(131,392)(132,391)(133,390)(134,389)(135,388)(136,406)
(137,408)(138,407)(139,414)(140,413)(141,412)(142,411)(143,410)(144,409)
(145,432)(146,431)(147,430)(148,429)(149,428)(150,427)(151,426)(152,425)
(153,424)(154,423)(155,422)(156,421)(157,420)(158,419)(159,418)(160,417)
(161,416)(162,415)(163,325)(164,327)(165,326)(166,333)(167,332)(168,331)
(169,330)(170,329)(171,328)(172,351)(173,350)(174,349)(175,348)(176,347)
(177,346)(178,345)(179,344)(180,343)(181,342)(182,341)(183,340)(184,339)
(185,338)(186,337)(187,336)(188,335)(189,334)(190,352)(191,354)(192,353)
(193,360)(194,359)(195,358)(196,357)(197,356)(198,355)(199,378)(200,377)
(201,376)(202,375)(203,374)(204,373)(205,372)(206,371)(207,370)(208,369)
(209,368)(210,367)(211,366)(212,365)(213,364)(214,363)(215,362)(216,361);;
s2 := ( 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, 64)( 56, 66)( 57, 65)( 58, 72)( 59, 71)( 60, 70)
( 61, 69)( 62, 68)( 63, 67)( 73, 81)( 74, 80)( 75, 79)( 76, 78)( 82, 91)
( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)( 90, 94)
(100,108)(101,107)(102,106)(103,105)(109,118)(110,120)(111,119)(112,126)
(113,125)(114,124)(115,123)(116,122)(117,121)(127,135)(128,134)(129,133)
(130,132)(136,145)(137,147)(138,146)(139,153)(140,152)(141,151)(142,150)
(143,149)(144,148)(154,162)(155,161)(156,160)(157,159)(163,172)(164,174)
(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)(181,189)
(182,188)(183,187)(184,186)(190,199)(191,201)(192,200)(193,207)(194,206)
(195,205)(196,204)(197,203)(198,202)(208,216)(209,215)(210,214)(211,213)
(217,226)(218,228)(219,227)(220,234)(221,233)(222,232)(223,231)(224,230)
(225,229)(235,243)(236,242)(237,241)(238,240)(244,253)(245,255)(246,254)
(247,261)(248,260)(249,259)(250,258)(251,257)(252,256)(262,270)(263,269)
(264,268)(265,267)(271,280)(272,282)(273,281)(274,288)(275,287)(276,286)
(277,285)(278,284)(279,283)(289,297)(290,296)(291,295)(292,294)(298,307)
(299,309)(300,308)(301,315)(302,314)(303,313)(304,312)(305,311)(306,310)
(316,324)(317,323)(318,322)(319,321)(325,334)(326,336)(327,335)(328,342)
(329,341)(330,340)(331,339)(332,338)(333,337)(343,351)(344,350)(345,349)
(346,348)(352,361)(353,363)(354,362)(355,369)(356,368)(357,367)(358,366)
(359,365)(360,364)(370,378)(371,377)(372,376)(373,375)(379,388)(380,390)
(381,389)(382,396)(383,395)(384,394)(385,393)(386,392)(387,391)(397,405)
(398,404)(399,403)(400,402)(406,415)(407,417)(408,416)(409,423)(410,422)
(411,421)(412,420)(413,419)(414,418)(424,432)(425,431)(426,430)(427,429);;
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, s0*s1*s2*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,
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)!( 55, 82)( 56, 83)( 57, 84)( 58, 85)( 59, 86)( 60, 87)( 61, 88)
( 62, 89)( 63, 90)( 64, 91)( 65, 92)( 66, 93)( 67, 94)( 68, 95)( 69, 96)
( 70, 97)( 71, 98)( 72, 99)( 73,100)( 74,101)( 75,102)( 76,103)( 77,104)
( 78,105)( 79,106)( 80,107)( 81,108)(109,163)(110,164)(111,165)(112,166)
(113,167)(114,168)(115,169)(116,170)(117,171)(118,172)(119,173)(120,174)
(121,175)(122,176)(123,177)(124,178)(125,179)(126,180)(127,181)(128,182)
(129,183)(130,184)(131,185)(132,186)(133,187)(134,188)(135,189)(136,190)
(137,191)(138,192)(139,193)(140,194)(141,195)(142,196)(143,197)(144,198)
(145,199)(146,200)(147,201)(148,202)(149,203)(150,204)(151,205)(152,206)
(153,207)(154,208)(155,209)(156,210)(157,211)(158,212)(159,213)(160,214)
(161,215)(162,216)(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)
(223,331)(224,332)(225,333)(226,334)(227,335)(228,336)(229,337)(230,338)
(231,339)(232,340)(233,341)(234,342)(235,343)(236,344)(237,345)(238,346)
(239,347)(240,348)(241,349)(242,350)(243,351)(244,352)(245,353)(246,354)
(247,355)(248,356)(249,357)(250,358)(251,359)(252,360)(253,361)(254,362)
(255,363)(256,364)(257,365)(258,366)(259,367)(260,368)(261,369)(262,370)
(263,371)(264,372)(265,373)(266,374)(267,375)(268,376)(269,377)(270,378)
(271,406)(272,407)(273,408)(274,409)(275,410)(276,411)(277,412)(278,413)
(279,414)(280,415)(281,416)(282,417)(283,418)(284,419)(285,420)(286,421)
(287,422)(288,423)(289,424)(290,425)(291,426)(292,427)(293,428)(294,429)
(295,430)(296,431)(297,432)(298,379)(299,380)(300,381)(301,382)(302,383)
(303,384)(304,385)(305,386)(306,387)(307,388)(308,389)(309,390)(310,391)
(311,392)(312,393)(313,394)(314,395)(315,396)(316,397)(317,398)(318,399)
(319,400)(320,401)(321,402)(322,403)(323,404)(324,405);
s1 := Sym(432)!( 1,217)( 2,219)( 3,218)( 4,225)( 5,224)( 6,223)( 7,222)
( 8,221)( 9,220)( 10,243)( 11,242)( 12,241)( 13,240)( 14,239)( 15,238)
( 16,237)( 17,236)( 18,235)( 19,234)( 20,233)( 21,232)( 22,231)( 23,230)
( 24,229)( 25,228)( 26,227)( 27,226)( 28,244)( 29,246)( 30,245)( 31,252)
( 32,251)( 33,250)( 34,249)( 35,248)( 36,247)( 37,270)( 38,269)( 39,268)
( 40,267)( 41,266)( 42,265)( 43,264)( 44,263)( 45,262)( 46,261)( 47,260)
( 48,259)( 49,258)( 50,257)( 51,256)( 52,255)( 53,254)( 54,253)( 55,298)
( 56,300)( 57,299)( 58,306)( 59,305)( 60,304)( 61,303)( 62,302)( 63,301)
( 64,324)( 65,323)( 66,322)( 67,321)( 68,320)( 69,319)( 70,318)( 71,317)
( 72,316)( 73,315)( 74,314)( 75,313)( 76,312)( 77,311)( 78,310)( 79,309)
( 80,308)( 81,307)( 82,271)( 83,273)( 84,272)( 85,279)( 86,278)( 87,277)
( 88,276)( 89,275)( 90,274)( 91,297)( 92,296)( 93,295)( 94,294)( 95,293)
( 96,292)( 97,291)( 98,290)( 99,289)(100,288)(101,287)(102,286)(103,285)
(104,284)(105,283)(106,282)(107,281)(108,280)(109,379)(110,381)(111,380)
(112,387)(113,386)(114,385)(115,384)(116,383)(117,382)(118,405)(119,404)
(120,403)(121,402)(122,401)(123,400)(124,399)(125,398)(126,397)(127,396)
(128,395)(129,394)(130,393)(131,392)(132,391)(133,390)(134,389)(135,388)
(136,406)(137,408)(138,407)(139,414)(140,413)(141,412)(142,411)(143,410)
(144,409)(145,432)(146,431)(147,430)(148,429)(149,428)(150,427)(151,426)
(152,425)(153,424)(154,423)(155,422)(156,421)(157,420)(158,419)(159,418)
(160,417)(161,416)(162,415)(163,325)(164,327)(165,326)(166,333)(167,332)
(168,331)(169,330)(170,329)(171,328)(172,351)(173,350)(174,349)(175,348)
(176,347)(177,346)(178,345)(179,344)(180,343)(181,342)(182,341)(183,340)
(184,339)(185,338)(186,337)(187,336)(188,335)(189,334)(190,352)(191,354)
(192,353)(193,360)(194,359)(195,358)(196,357)(197,356)(198,355)(199,378)
(200,377)(201,376)(202,375)(203,374)(204,373)(205,372)(206,371)(207,370)
(208,369)(209,368)(210,367)(211,366)(212,365)(213,364)(214,363)(215,362)
(216,361);
s2 := 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, 64)( 56, 66)( 57, 65)( 58, 72)( 59, 71)
( 60, 70)( 61, 69)( 62, 68)( 63, 67)( 73, 81)( 74, 80)( 75, 79)( 76, 78)
( 82, 91)( 83, 93)( 84, 92)( 85, 99)( 86, 98)( 87, 97)( 88, 96)( 89, 95)
( 90, 94)(100,108)(101,107)(102,106)(103,105)(109,118)(110,120)(111,119)
(112,126)(113,125)(114,124)(115,123)(116,122)(117,121)(127,135)(128,134)
(129,133)(130,132)(136,145)(137,147)(138,146)(139,153)(140,152)(141,151)
(142,150)(143,149)(144,148)(154,162)(155,161)(156,160)(157,159)(163,172)
(164,174)(165,173)(166,180)(167,179)(168,178)(169,177)(170,176)(171,175)
(181,189)(182,188)(183,187)(184,186)(190,199)(191,201)(192,200)(193,207)
(194,206)(195,205)(196,204)(197,203)(198,202)(208,216)(209,215)(210,214)
(211,213)(217,226)(218,228)(219,227)(220,234)(221,233)(222,232)(223,231)
(224,230)(225,229)(235,243)(236,242)(237,241)(238,240)(244,253)(245,255)
(246,254)(247,261)(248,260)(249,259)(250,258)(251,257)(252,256)(262,270)
(263,269)(264,268)(265,267)(271,280)(272,282)(273,281)(274,288)(275,287)
(276,286)(277,285)(278,284)(279,283)(289,297)(290,296)(291,295)(292,294)
(298,307)(299,309)(300,308)(301,315)(302,314)(303,313)(304,312)(305,311)
(306,310)(316,324)(317,323)(318,322)(319,321)(325,334)(326,336)(327,335)
(328,342)(329,341)(330,340)(331,339)(332,338)(333,337)(343,351)(344,350)
(345,349)(346,348)(352,361)(353,363)(354,362)(355,369)(356,368)(357,367)
(358,366)(359,365)(360,364)(370,378)(371,377)(372,376)(373,375)(379,388)
(380,390)(381,389)(382,396)(383,395)(384,394)(385,393)(386,392)(387,391)
(397,405)(398,404)(399,403)(400,402)(406,415)(407,417)(408,416)(409,423)
(410,422)(411,421)(412,420)(413,419)(414,418)(424,432)(425,431)(426,430)
(427,429);
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, s0*s1*s2*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,
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