Polytope of Type {54,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,4}*1728a
if this polytope has a name.
Group : SmallGroup(1728,2197)
Rank : 3
Schlafli Type : {54,4}
Number of vertices, edges, etc : 216, 432, 16
Order of s0s1s2 : 54
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {18,4}*576a
4-fold quotients : {54,4}*432b
8-fold quotients : {27,4}*216
9-fold quotients : {6,4}*192a
12-fold quotients : {18,4}*144b
24-fold quotients : {9,4}*72
36-fold quotients : {6,4}*48c
72-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)( 26, 46)
( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49,129)( 50,130)
( 51,132)( 52,131)( 53,133)( 54,134)( 55,136)( 56,135)( 57,141)( 58,142)
( 59,144)( 60,143)( 61,137)( 62,138)( 63,140)( 64,139)( 65,113)( 66,114)
( 67,116)( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,125)( 74,126)
( 75,128)( 76,127)( 77,121)( 78,122)( 79,124)( 80,123)( 81, 97)( 82, 98)
( 83,100)( 84, 99)( 85,101)( 86,102)( 87,104)( 88,103)( 89,109)( 90,110)
( 91,112)( 92,111)( 93,105)( 94,106)( 95,108)( 96,107)(145,417)(146,418)
(147,420)(148,419)(149,421)(150,422)(151,424)(152,423)(153,429)(154,430)
(155,432)(156,431)(157,425)(158,426)(159,428)(160,427)(161,401)(162,402)
(163,404)(164,403)(165,405)(166,406)(167,408)(168,407)(169,413)(170,414)
(171,416)(172,415)(173,409)(174,410)(175,412)(176,411)(177,385)(178,386)
(179,388)(180,387)(181,389)(182,390)(183,392)(184,391)(185,397)(186,398)
(187,400)(188,399)(189,393)(190,394)(191,396)(192,395)(193,369)(194,370)
(195,372)(196,371)(197,373)(198,374)(199,376)(200,375)(201,381)(202,382)
(203,384)(204,383)(205,377)(206,378)(207,380)(208,379)(209,353)(210,354)
(211,356)(212,355)(213,357)(214,358)(215,360)(216,359)(217,365)(218,366)
(219,368)(220,367)(221,361)(222,362)(223,364)(224,363)(225,337)(226,338)
(227,340)(228,339)(229,341)(230,342)(231,344)(232,343)(233,349)(234,350)
(235,352)(236,351)(237,345)(238,346)(239,348)(240,347)(241,321)(242,322)
(243,324)(244,323)(245,325)(246,326)(247,328)(248,327)(249,333)(250,334)
(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,305)(258,306)
(259,308)(260,307)(261,309)(262,310)(263,312)(264,311)(265,317)(266,318)
(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,289)(274,290)
(275,292)(276,291)(277,293)(278,294)(279,296)(280,295)(281,301)(282,302)
(283,304)(284,303)(285,297)(286,298)(287,300)(288,299);;
s1 := ( 1,145)( 2,148)( 3,147)( 4,146)( 5,160)( 6,157)( 7,158)( 8,159)
( 9,155)( 10,154)( 11,153)( 12,156)( 13,150)( 14,151)( 15,152)( 16,149)
( 17,177)( 18,180)( 19,179)( 20,178)( 21,192)( 22,189)( 23,190)( 24,191)
( 25,187)( 26,186)( 27,185)( 28,188)( 29,182)( 30,183)( 31,184)( 32,181)
( 33,161)( 34,164)( 35,163)( 36,162)( 37,176)( 38,173)( 39,174)( 40,175)
( 41,171)( 42,170)( 43,169)( 44,172)( 45,166)( 46,167)( 47,168)( 48,165)
( 49,273)( 50,276)( 51,275)( 52,274)( 53,288)( 54,285)( 55,286)( 56,287)
( 57,283)( 58,282)( 59,281)( 60,284)( 61,278)( 62,279)( 63,280)( 64,277)
( 65,257)( 66,260)( 67,259)( 68,258)( 69,272)( 70,269)( 71,270)( 72,271)
( 73,267)( 74,266)( 75,265)( 76,268)( 77,262)( 78,263)( 79,264)( 80,261)
( 81,241)( 82,244)( 83,243)( 84,242)( 85,256)( 86,253)( 87,254)( 88,255)
( 89,251)( 90,250)( 91,249)( 92,252)( 93,246)( 94,247)( 95,248)( 96,245)
( 97,225)( 98,228)( 99,227)(100,226)(101,240)(102,237)(103,238)(104,239)
(105,235)(106,234)(107,233)(108,236)(109,230)(110,231)(111,232)(112,229)
(113,209)(114,212)(115,211)(116,210)(117,224)(118,221)(119,222)(120,223)
(121,219)(122,218)(123,217)(124,220)(125,214)(126,215)(127,216)(128,213)
(129,193)(130,196)(131,195)(132,194)(133,208)(134,205)(135,206)(136,207)
(137,203)(138,202)(139,201)(140,204)(141,198)(142,199)(143,200)(144,197)
(289,417)(290,420)(291,419)(292,418)(293,432)(294,429)(295,430)(296,431)
(297,427)(298,426)(299,425)(300,428)(301,422)(302,423)(303,424)(304,421)
(305,401)(306,404)(307,403)(308,402)(309,416)(310,413)(311,414)(312,415)
(313,411)(314,410)(315,409)(316,412)(317,406)(318,407)(319,408)(320,405)
(321,385)(322,388)(323,387)(324,386)(325,400)(326,397)(327,398)(328,399)
(329,395)(330,394)(331,393)(332,396)(333,390)(334,391)(335,392)(336,389)
(337,369)(338,372)(339,371)(340,370)(341,384)(342,381)(343,382)(344,383)
(345,379)(346,378)(347,377)(348,380)(349,374)(350,375)(351,376)(352,373)
(354,356)(357,368)(358,365)(359,366)(360,367)(361,363);;
s2 := ( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)( 76, 80)
( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)(108,112)
(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)(124,128)
(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)(140,144)
(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)(156,160)
(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)(172,176)
(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)(188,192)
(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)(204,208)
(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)(220,224)
(225,229)(226,230)(227,231)(228,232)(233,237)(234,238)(235,239)(236,240)
(241,245)(242,246)(243,247)(244,248)(249,253)(250,254)(251,255)(252,256)
(257,261)(258,262)(259,263)(260,264)(265,269)(266,270)(267,271)(268,272)
(273,277)(274,278)(275,279)(276,280)(281,285)(282,286)(283,287)(284,288)
(289,293)(290,294)(291,295)(292,296)(297,301)(298,302)(299,303)(300,304)
(305,309)(306,310)(307,311)(308,312)(313,317)(314,318)(315,319)(316,320)
(321,325)(322,326)(323,327)(324,328)(329,333)(330,334)(331,335)(332,336)
(337,341)(338,342)(339,343)(340,344)(345,349)(346,350)(347,351)(348,352)
(353,357)(354,358)(355,359)(356,360)(361,365)(362,366)(363,367)(364,368)
(369,373)(370,374)(371,375)(372,376)(377,381)(378,382)(379,383)(380,384)
(385,389)(386,390)(387,391)(388,392)(393,397)(394,398)(395,399)(396,400)
(401,405)(402,406)(403,407)(404,408)(409,413)(410,414)(411,415)(412,416)
(417,421)(418,422)(419,423)(420,424)(425,429)(426,430)(427,431)(428,432);;
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*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)!( 3, 4)( 7, 8)( 9, 13)( 10, 14)( 11, 16)( 12, 15)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 37)( 22, 38)( 23, 40)( 24, 39)( 25, 45)
( 26, 46)( 27, 48)( 28, 47)( 29, 41)( 30, 42)( 31, 44)( 32, 43)( 49,129)
( 50,130)( 51,132)( 52,131)( 53,133)( 54,134)( 55,136)( 56,135)( 57,141)
( 58,142)( 59,144)( 60,143)( 61,137)( 62,138)( 63,140)( 64,139)( 65,113)
( 66,114)( 67,116)( 68,115)( 69,117)( 70,118)( 71,120)( 72,119)( 73,125)
( 74,126)( 75,128)( 76,127)( 77,121)( 78,122)( 79,124)( 80,123)( 81, 97)
( 82, 98)( 83,100)( 84, 99)( 85,101)( 86,102)( 87,104)( 88,103)( 89,109)
( 90,110)( 91,112)( 92,111)( 93,105)( 94,106)( 95,108)( 96,107)(145,417)
(146,418)(147,420)(148,419)(149,421)(150,422)(151,424)(152,423)(153,429)
(154,430)(155,432)(156,431)(157,425)(158,426)(159,428)(160,427)(161,401)
(162,402)(163,404)(164,403)(165,405)(166,406)(167,408)(168,407)(169,413)
(170,414)(171,416)(172,415)(173,409)(174,410)(175,412)(176,411)(177,385)
(178,386)(179,388)(180,387)(181,389)(182,390)(183,392)(184,391)(185,397)
(186,398)(187,400)(188,399)(189,393)(190,394)(191,396)(192,395)(193,369)
(194,370)(195,372)(196,371)(197,373)(198,374)(199,376)(200,375)(201,381)
(202,382)(203,384)(204,383)(205,377)(206,378)(207,380)(208,379)(209,353)
(210,354)(211,356)(212,355)(213,357)(214,358)(215,360)(216,359)(217,365)
(218,366)(219,368)(220,367)(221,361)(222,362)(223,364)(224,363)(225,337)
(226,338)(227,340)(228,339)(229,341)(230,342)(231,344)(232,343)(233,349)
(234,350)(235,352)(236,351)(237,345)(238,346)(239,348)(240,347)(241,321)
(242,322)(243,324)(244,323)(245,325)(246,326)(247,328)(248,327)(249,333)
(250,334)(251,336)(252,335)(253,329)(254,330)(255,332)(256,331)(257,305)
(258,306)(259,308)(260,307)(261,309)(262,310)(263,312)(264,311)(265,317)
(266,318)(267,320)(268,319)(269,313)(270,314)(271,316)(272,315)(273,289)
(274,290)(275,292)(276,291)(277,293)(278,294)(279,296)(280,295)(281,301)
(282,302)(283,304)(284,303)(285,297)(286,298)(287,300)(288,299);
s1 := Sym(432)!( 1,145)( 2,148)( 3,147)( 4,146)( 5,160)( 6,157)( 7,158)
( 8,159)( 9,155)( 10,154)( 11,153)( 12,156)( 13,150)( 14,151)( 15,152)
( 16,149)( 17,177)( 18,180)( 19,179)( 20,178)( 21,192)( 22,189)( 23,190)
( 24,191)( 25,187)( 26,186)( 27,185)( 28,188)( 29,182)( 30,183)( 31,184)
( 32,181)( 33,161)( 34,164)( 35,163)( 36,162)( 37,176)( 38,173)( 39,174)
( 40,175)( 41,171)( 42,170)( 43,169)( 44,172)( 45,166)( 46,167)( 47,168)
( 48,165)( 49,273)( 50,276)( 51,275)( 52,274)( 53,288)( 54,285)( 55,286)
( 56,287)( 57,283)( 58,282)( 59,281)( 60,284)( 61,278)( 62,279)( 63,280)
( 64,277)( 65,257)( 66,260)( 67,259)( 68,258)( 69,272)( 70,269)( 71,270)
( 72,271)( 73,267)( 74,266)( 75,265)( 76,268)( 77,262)( 78,263)( 79,264)
( 80,261)( 81,241)( 82,244)( 83,243)( 84,242)( 85,256)( 86,253)( 87,254)
( 88,255)( 89,251)( 90,250)( 91,249)( 92,252)( 93,246)( 94,247)( 95,248)
( 96,245)( 97,225)( 98,228)( 99,227)(100,226)(101,240)(102,237)(103,238)
(104,239)(105,235)(106,234)(107,233)(108,236)(109,230)(110,231)(111,232)
(112,229)(113,209)(114,212)(115,211)(116,210)(117,224)(118,221)(119,222)
(120,223)(121,219)(122,218)(123,217)(124,220)(125,214)(126,215)(127,216)
(128,213)(129,193)(130,196)(131,195)(132,194)(133,208)(134,205)(135,206)
(136,207)(137,203)(138,202)(139,201)(140,204)(141,198)(142,199)(143,200)
(144,197)(289,417)(290,420)(291,419)(292,418)(293,432)(294,429)(295,430)
(296,431)(297,427)(298,426)(299,425)(300,428)(301,422)(302,423)(303,424)
(304,421)(305,401)(306,404)(307,403)(308,402)(309,416)(310,413)(311,414)
(312,415)(313,411)(314,410)(315,409)(316,412)(317,406)(318,407)(319,408)
(320,405)(321,385)(322,388)(323,387)(324,386)(325,400)(326,397)(327,398)
(328,399)(329,395)(330,394)(331,393)(332,396)(333,390)(334,391)(335,392)
(336,389)(337,369)(338,372)(339,371)(340,370)(341,384)(342,381)(343,382)
(344,383)(345,379)(346,378)(347,377)(348,380)(349,374)(350,375)(351,376)
(352,373)(354,356)(357,368)(358,365)(359,366)(360,367)(361,363);
s2 := Sym(432)!( 1, 5)( 2, 6)( 3, 7)( 4, 8)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 17, 21)( 18, 22)( 19, 23)( 20, 24)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 33, 37)( 34, 38)( 35, 39)( 36, 40)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 49, 53)( 50, 54)( 51, 55)( 52, 56)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 69)( 66, 70)( 67, 71)( 68, 72)( 73, 77)( 74, 78)( 75, 79)
( 76, 80)( 81, 85)( 82, 86)( 83, 87)( 84, 88)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)( 97,101)( 98,102)( 99,103)(100,104)(105,109)(106,110)(107,111)
(108,112)(113,117)(114,118)(115,119)(116,120)(121,125)(122,126)(123,127)
(124,128)(129,133)(130,134)(131,135)(132,136)(137,141)(138,142)(139,143)
(140,144)(145,149)(146,150)(147,151)(148,152)(153,157)(154,158)(155,159)
(156,160)(161,165)(162,166)(163,167)(164,168)(169,173)(170,174)(171,175)
(172,176)(177,181)(178,182)(179,183)(180,184)(185,189)(186,190)(187,191)
(188,192)(193,197)(194,198)(195,199)(196,200)(201,205)(202,206)(203,207)
(204,208)(209,213)(210,214)(211,215)(212,216)(217,221)(218,222)(219,223)
(220,224)(225,229)(226,230)(227,231)(228,232)(233,237)(234,238)(235,239)
(236,240)(241,245)(242,246)(243,247)(244,248)(249,253)(250,254)(251,255)
(252,256)(257,261)(258,262)(259,263)(260,264)(265,269)(266,270)(267,271)
(268,272)(273,277)(274,278)(275,279)(276,280)(281,285)(282,286)(283,287)
(284,288)(289,293)(290,294)(291,295)(292,296)(297,301)(298,302)(299,303)
(300,304)(305,309)(306,310)(307,311)(308,312)(313,317)(314,318)(315,319)
(316,320)(321,325)(322,326)(323,327)(324,328)(329,333)(330,334)(331,335)
(332,336)(337,341)(338,342)(339,343)(340,344)(345,349)(346,350)(347,351)
(348,352)(353,357)(354,358)(355,359)(356,360)(361,365)(362,366)(363,367)
(364,368)(369,373)(370,374)(371,375)(372,376)(377,381)(378,382)(379,383)
(380,384)(385,389)(386,390)(387,391)(388,392)(393,397)(394,398)(395,399)
(396,400)(401,405)(402,406)(403,407)(404,408)(409,413)(410,414)(411,415)
(412,416)(417,421)(418,422)(419,423)(420,424)(425,429)(426,430)(427,431)
(428,432);
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, s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1*s0*s1*s2*s1,
s0*s1*s0*s1*s2*s1*s0*s1*s0*s1*s0*s1*s2*s1*s0*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