Polytope of Type {4,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,54}*1728b
if this polytope has a name.
Group : SmallGroup(1728,11358)
Rank : 3
Schlafli Type : {4,54}
Number of vertices, edges, etc : 16, 432, 216
Order of s0s1s2 : 108
Order of s0s1s2s1 : 4
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 : {4,18}*576b
4-fold quotients : {4,54}*432a, {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 : {4,6}*192b
12-fold quotients : {4,18}*144a, {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 : {4,6}*48a, {4,3}*48, {4,6}*48b, {4,6}*48c
48-fold quotients : {2,9}*36
72-fold quotients : {4,3}*24, {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 := ( 1, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)( 14, 16)
( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)( 30, 32)
( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)( 46, 48)
( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)( 62, 64)
( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)( 78, 80)
( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)( 94, 96)
( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)(110,112)
(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)(126,128)
(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)(142,144)
(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)(158,160)
(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)(174,176)
(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)(190,192)
(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)(206,208)
(209,211)(210,212)(213,215)(214,216)(217,327)(218,328)(219,325)(220,326)
(221,331)(222,332)(223,329)(224,330)(225,335)(226,336)(227,333)(228,334)
(229,339)(230,340)(231,337)(232,338)(233,343)(234,344)(235,341)(236,342)
(237,347)(238,348)(239,345)(240,346)(241,351)(242,352)(243,349)(244,350)
(245,355)(246,356)(247,353)(248,354)(249,359)(250,360)(251,357)(252,358)
(253,363)(254,364)(255,361)(256,362)(257,367)(258,368)(259,365)(260,366)
(261,371)(262,372)(263,369)(264,370)(265,375)(266,376)(267,373)(268,374)
(269,379)(270,380)(271,377)(272,378)(273,383)(274,384)(275,381)(276,382)
(277,387)(278,388)(279,385)(280,386)(281,391)(282,392)(283,389)(284,390)
(285,395)(286,396)(287,393)(288,394)(289,399)(290,400)(291,397)(292,398)
(293,403)(294,404)(295,401)(296,402)(297,407)(298,408)(299,405)(300,406)
(301,411)(302,412)(303,409)(304,410)(305,415)(306,416)(307,413)(308,414)
(309,419)(310,420)(311,417)(312,418)(313,423)(314,424)(315,421)(316,422)
(317,427)(318,428)(319,425)(320,426)(321,431)(322,432)(323,429)(324,430);;
s1 := ( 1,217)( 2,218)( 3,220)( 4,219)( 5,225)( 6,226)( 7,228)( 8,227)
( 9,221)( 10,222)( 11,224)( 12,223)( 13,249)( 14,250)( 15,252)( 16,251)
( 17,245)( 18,246)( 19,248)( 20,247)( 21,241)( 22,242)( 23,244)( 24,243)
( 25,237)( 26,238)( 27,240)( 28,239)( 29,233)( 30,234)( 31,236)( 32,235)
( 33,229)( 34,230)( 35,232)( 36,231)( 37,321)( 38,322)( 39,324)( 40,323)
( 41,317)( 42,318)( 43,320)( 44,319)( 45,313)( 46,314)( 47,316)( 48,315)
( 49,309)( 50,310)( 51,312)( 52,311)( 53,305)( 54,306)( 55,308)( 56,307)
( 57,301)( 58,302)( 59,304)( 60,303)( 61,297)( 62,298)( 63,300)( 64,299)
( 65,293)( 66,294)( 67,296)( 68,295)( 69,289)( 70,290)( 71,292)( 72,291)
( 73,285)( 74,286)( 75,288)( 76,287)( 77,281)( 78,282)( 79,284)( 80,283)
( 81,277)( 82,278)( 83,280)( 84,279)( 85,273)( 86,274)( 87,276)( 88,275)
( 89,269)( 90,270)( 91,272)( 92,271)( 93,265)( 94,266)( 95,268)( 96,267)
( 97,261)( 98,262)( 99,264)(100,263)(101,257)(102,258)(103,260)(104,259)
(105,253)(106,254)(107,256)(108,255)(109,325)(110,326)(111,328)(112,327)
(113,333)(114,334)(115,336)(116,335)(117,329)(118,330)(119,332)(120,331)
(121,357)(122,358)(123,360)(124,359)(125,353)(126,354)(127,356)(128,355)
(129,349)(130,350)(131,352)(132,351)(133,345)(134,346)(135,348)(136,347)
(137,341)(138,342)(139,344)(140,343)(141,337)(142,338)(143,340)(144,339)
(145,429)(146,430)(147,432)(148,431)(149,425)(150,426)(151,428)(152,427)
(153,421)(154,422)(155,424)(156,423)(157,417)(158,418)(159,420)(160,419)
(161,413)(162,414)(163,416)(164,415)(165,409)(166,410)(167,412)(168,411)
(169,405)(170,406)(171,408)(172,407)(173,401)(174,402)(175,404)(176,403)
(177,397)(178,398)(179,400)(180,399)(181,393)(182,394)(183,396)(184,395)
(185,389)(186,390)(187,392)(188,391)(189,385)(190,386)(191,388)(192,387)
(193,381)(194,382)(195,384)(196,383)(197,377)(198,378)(199,380)(200,379)
(201,373)(202,374)(203,376)(204,375)(205,369)(206,370)(207,372)(208,371)
(209,365)(210,366)(211,368)(212,367)(213,361)(214,362)(215,364)(216,363);;
s2 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)( 8, 46)
( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 69)( 14, 72)( 15, 71)( 16, 70)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 61)( 22, 64)( 23, 63)( 24, 62)
( 25, 57)( 26, 60)( 27, 59)( 28, 58)( 29, 53)( 30, 56)( 31, 55)( 32, 54)
( 33, 49)( 34, 52)( 35, 51)( 36, 50)( 73,105)( 74,108)( 75,107)( 76,106)
( 77,101)( 78,104)( 79,103)( 80,102)( 81, 97)( 82,100)( 83, 99)( 84, 98)
( 85, 93)( 86, 96)( 87, 95)( 88, 94)( 90, 92)(109,145)(110,148)(111,147)
(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)
(120,150)(121,177)(122,180)(123,179)(124,178)(125,173)(126,176)(127,175)
(128,174)(129,169)(130,172)(131,171)(132,170)(133,165)(134,168)(135,167)
(136,166)(137,161)(138,164)(139,163)(140,162)(141,157)(142,160)(143,159)
(144,158)(181,213)(182,216)(183,215)(184,214)(185,209)(186,212)(187,211)
(188,210)(189,205)(190,208)(191,207)(192,206)(193,201)(194,204)(195,203)
(196,202)(198,200)(217,253)(218,256)(219,255)(220,254)(221,261)(222,264)
(223,263)(224,262)(225,257)(226,260)(227,259)(228,258)(229,285)(230,288)
(231,287)(232,286)(233,281)(234,284)(235,283)(236,282)(237,277)(238,280)
(239,279)(240,278)(241,273)(242,276)(243,275)(244,274)(245,269)(246,272)
(247,271)(248,270)(249,265)(250,268)(251,267)(252,266)(289,321)(290,324)
(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)(297,313)(298,316)
(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)(306,308)(325,361)
(326,364)(327,363)(328,362)(329,369)(330,372)(331,371)(332,370)(333,365)
(334,368)(335,367)(336,366)(337,393)(338,396)(339,395)(340,394)(341,389)
(342,392)(343,391)(344,390)(345,385)(346,388)(347,387)(348,386)(349,381)
(350,384)(351,383)(352,382)(353,377)(354,380)(355,379)(356,378)(357,373)
(358,376)(359,375)(360,374)(397,429)(398,432)(399,431)(400,430)(401,425)
(402,428)(403,427)(404,426)(405,421)(406,424)(407,423)(408,422)(409,417)
(410,420)(411,419)(412,418)(414,416);;
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*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, 3)( 2, 4)( 5, 7)( 6, 8)( 9, 11)( 10, 12)( 13, 15)
( 14, 16)( 17, 19)( 18, 20)( 21, 23)( 22, 24)( 25, 27)( 26, 28)( 29, 31)
( 30, 32)( 33, 35)( 34, 36)( 37, 39)( 38, 40)( 41, 43)( 42, 44)( 45, 47)
( 46, 48)( 49, 51)( 50, 52)( 53, 55)( 54, 56)( 57, 59)( 58, 60)( 61, 63)
( 62, 64)( 65, 67)( 66, 68)( 69, 71)( 70, 72)( 73, 75)( 74, 76)( 77, 79)
( 78, 80)( 81, 83)( 82, 84)( 85, 87)( 86, 88)( 89, 91)( 90, 92)( 93, 95)
( 94, 96)( 97, 99)( 98,100)(101,103)(102,104)(105,107)(106,108)(109,111)
(110,112)(113,115)(114,116)(117,119)(118,120)(121,123)(122,124)(125,127)
(126,128)(129,131)(130,132)(133,135)(134,136)(137,139)(138,140)(141,143)
(142,144)(145,147)(146,148)(149,151)(150,152)(153,155)(154,156)(157,159)
(158,160)(161,163)(162,164)(165,167)(166,168)(169,171)(170,172)(173,175)
(174,176)(177,179)(178,180)(181,183)(182,184)(185,187)(186,188)(189,191)
(190,192)(193,195)(194,196)(197,199)(198,200)(201,203)(202,204)(205,207)
(206,208)(209,211)(210,212)(213,215)(214,216)(217,327)(218,328)(219,325)
(220,326)(221,331)(222,332)(223,329)(224,330)(225,335)(226,336)(227,333)
(228,334)(229,339)(230,340)(231,337)(232,338)(233,343)(234,344)(235,341)
(236,342)(237,347)(238,348)(239,345)(240,346)(241,351)(242,352)(243,349)
(244,350)(245,355)(246,356)(247,353)(248,354)(249,359)(250,360)(251,357)
(252,358)(253,363)(254,364)(255,361)(256,362)(257,367)(258,368)(259,365)
(260,366)(261,371)(262,372)(263,369)(264,370)(265,375)(266,376)(267,373)
(268,374)(269,379)(270,380)(271,377)(272,378)(273,383)(274,384)(275,381)
(276,382)(277,387)(278,388)(279,385)(280,386)(281,391)(282,392)(283,389)
(284,390)(285,395)(286,396)(287,393)(288,394)(289,399)(290,400)(291,397)
(292,398)(293,403)(294,404)(295,401)(296,402)(297,407)(298,408)(299,405)
(300,406)(301,411)(302,412)(303,409)(304,410)(305,415)(306,416)(307,413)
(308,414)(309,419)(310,420)(311,417)(312,418)(313,423)(314,424)(315,421)
(316,422)(317,427)(318,428)(319,425)(320,426)(321,431)(322,432)(323,429)
(324,430);
s1 := Sym(432)!( 1,217)( 2,218)( 3,220)( 4,219)( 5,225)( 6,226)( 7,228)
( 8,227)( 9,221)( 10,222)( 11,224)( 12,223)( 13,249)( 14,250)( 15,252)
( 16,251)( 17,245)( 18,246)( 19,248)( 20,247)( 21,241)( 22,242)( 23,244)
( 24,243)( 25,237)( 26,238)( 27,240)( 28,239)( 29,233)( 30,234)( 31,236)
( 32,235)( 33,229)( 34,230)( 35,232)( 36,231)( 37,321)( 38,322)( 39,324)
( 40,323)( 41,317)( 42,318)( 43,320)( 44,319)( 45,313)( 46,314)( 47,316)
( 48,315)( 49,309)( 50,310)( 51,312)( 52,311)( 53,305)( 54,306)( 55,308)
( 56,307)( 57,301)( 58,302)( 59,304)( 60,303)( 61,297)( 62,298)( 63,300)
( 64,299)( 65,293)( 66,294)( 67,296)( 68,295)( 69,289)( 70,290)( 71,292)
( 72,291)( 73,285)( 74,286)( 75,288)( 76,287)( 77,281)( 78,282)( 79,284)
( 80,283)( 81,277)( 82,278)( 83,280)( 84,279)( 85,273)( 86,274)( 87,276)
( 88,275)( 89,269)( 90,270)( 91,272)( 92,271)( 93,265)( 94,266)( 95,268)
( 96,267)( 97,261)( 98,262)( 99,264)(100,263)(101,257)(102,258)(103,260)
(104,259)(105,253)(106,254)(107,256)(108,255)(109,325)(110,326)(111,328)
(112,327)(113,333)(114,334)(115,336)(116,335)(117,329)(118,330)(119,332)
(120,331)(121,357)(122,358)(123,360)(124,359)(125,353)(126,354)(127,356)
(128,355)(129,349)(130,350)(131,352)(132,351)(133,345)(134,346)(135,348)
(136,347)(137,341)(138,342)(139,344)(140,343)(141,337)(142,338)(143,340)
(144,339)(145,429)(146,430)(147,432)(148,431)(149,425)(150,426)(151,428)
(152,427)(153,421)(154,422)(155,424)(156,423)(157,417)(158,418)(159,420)
(160,419)(161,413)(162,414)(163,416)(164,415)(165,409)(166,410)(167,412)
(168,411)(169,405)(170,406)(171,408)(172,407)(173,401)(174,402)(175,404)
(176,403)(177,397)(178,398)(179,400)(180,399)(181,393)(182,394)(183,396)
(184,395)(185,389)(186,390)(187,392)(188,391)(189,385)(190,386)(191,388)
(192,387)(193,381)(194,382)(195,384)(196,383)(197,377)(198,378)(199,380)
(200,379)(201,373)(202,374)(203,376)(204,375)(205,369)(206,370)(207,372)
(208,371)(209,365)(210,366)(211,368)(212,367)(213,361)(214,362)(215,364)
(216,363);
s2 := Sym(432)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)
( 8, 46)( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 69)( 14, 72)( 15, 71)
( 16, 70)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 61)( 22, 64)( 23, 63)
( 24, 62)( 25, 57)( 26, 60)( 27, 59)( 28, 58)( 29, 53)( 30, 56)( 31, 55)
( 32, 54)( 33, 49)( 34, 52)( 35, 51)( 36, 50)( 73,105)( 74,108)( 75,107)
( 76,106)( 77,101)( 78,104)( 79,103)( 80,102)( 81, 97)( 82,100)( 83, 99)
( 84, 98)( 85, 93)( 86, 96)( 87, 95)( 88, 94)( 90, 92)(109,145)(110,148)
(111,147)(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)
(119,151)(120,150)(121,177)(122,180)(123,179)(124,178)(125,173)(126,176)
(127,175)(128,174)(129,169)(130,172)(131,171)(132,170)(133,165)(134,168)
(135,167)(136,166)(137,161)(138,164)(139,163)(140,162)(141,157)(142,160)
(143,159)(144,158)(181,213)(182,216)(183,215)(184,214)(185,209)(186,212)
(187,211)(188,210)(189,205)(190,208)(191,207)(192,206)(193,201)(194,204)
(195,203)(196,202)(198,200)(217,253)(218,256)(219,255)(220,254)(221,261)
(222,264)(223,263)(224,262)(225,257)(226,260)(227,259)(228,258)(229,285)
(230,288)(231,287)(232,286)(233,281)(234,284)(235,283)(236,282)(237,277)
(238,280)(239,279)(240,278)(241,273)(242,276)(243,275)(244,274)(245,269)
(246,272)(247,271)(248,270)(249,265)(250,268)(251,267)(252,266)(289,321)
(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)(297,313)
(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)(306,308)
(325,361)(326,364)(327,363)(328,362)(329,369)(330,372)(331,371)(332,370)
(333,365)(334,368)(335,367)(336,366)(337,393)(338,396)(339,395)(340,394)
(341,389)(342,392)(343,391)(344,390)(345,385)(346,388)(347,387)(348,386)
(349,381)(350,384)(351,383)(352,382)(353,377)(354,380)(355,379)(356,378)
(357,373)(358,376)(359,375)(360,374)(397,429)(398,432)(399,431)(400,430)
(401,425)(402,428)(403,427)(404,426)(405,421)(406,424)(407,423)(408,422)
(409,417)(410,420)(411,419)(412,418)(414,416);
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*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s2*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1*s2*s1*s2*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