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