Polytope of Type {54,8}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,8}*1728a
if this polytope has a name.
Group : SmallGroup(1728,2198)
Rank : 3
Schlafli Type : {54,8}
Number of vertices, edges, etc : 108, 432, 16
Order of s0s1s2 : 27
Order of s0s1s2s1 : 8
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Non-Orientable
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
3-fold quotients : {18,8}*576a
4-fold quotients : {54,4}*432c
8-fold quotients : {27,4}*216
9-fold quotients : {6,8}*192a
12-fold quotients : {18,4}*144c
24-fold quotients : {9,4}*72
36-fold quotients : {6,4}*48b
72-fold quotients : {3,4}*24
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 6)( 9, 14)( 10, 13)( 11, 15)( 12, 16)( 17, 33)( 18, 34)
( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 46)( 26, 45)
( 27, 47)( 28, 48)( 29, 42)( 30, 41)( 31, 43)( 32, 44)( 49,129)( 50,130)
( 51,132)( 52,131)( 53,134)( 54,133)( 55,135)( 56,136)( 57,142)( 58,141)
( 59,143)( 60,144)( 61,138)( 62,137)( 63,139)( 64,140)( 65,113)( 66,114)
( 67,116)( 68,115)( 69,118)( 70,117)( 71,119)( 72,120)( 73,126)( 74,125)
( 75,127)( 76,128)( 77,122)( 78,121)( 79,123)( 80,124)( 81, 97)( 82, 98)
( 83,100)( 84, 99)( 85,102)( 86,101)( 87,103)( 88,104)( 89,110)( 90,109)
( 91,111)( 92,112)( 93,106)( 94,105)( 95,107)( 96,108)(145,417)(146,418)
(147,420)(148,419)(149,422)(150,421)(151,423)(152,424)(153,430)(154,429)
(155,431)(156,432)(157,426)(158,425)(159,427)(160,428)(161,401)(162,402)
(163,404)(164,403)(165,406)(166,405)(167,407)(168,408)(169,414)(170,413)
(171,415)(172,416)(173,410)(174,409)(175,411)(176,412)(177,385)(178,386)
(179,388)(180,387)(181,390)(182,389)(183,391)(184,392)(185,398)(186,397)
(187,399)(188,400)(189,394)(190,393)(191,395)(192,396)(193,369)(194,370)
(195,372)(196,371)(197,374)(198,373)(199,375)(200,376)(201,382)(202,381)
(203,383)(204,384)(205,378)(206,377)(207,379)(208,380)(209,353)(210,354)
(211,356)(212,355)(213,358)(214,357)(215,359)(216,360)(217,366)(218,365)
(219,367)(220,368)(221,362)(222,361)(223,363)(224,364)(225,337)(226,338)
(227,340)(228,339)(229,342)(230,341)(231,343)(232,344)(233,350)(234,349)
(235,351)(236,352)(237,346)(238,345)(239,347)(240,348)(241,321)(242,322)
(243,324)(244,323)(245,326)(246,325)(247,327)(248,328)(249,334)(250,333)
(251,335)(252,336)(253,330)(254,329)(255,331)(256,332)(257,305)(258,306)
(259,308)(260,307)(261,310)(262,309)(263,311)(264,312)(265,318)(266,317)
(267,319)(268,320)(269,314)(270,313)(271,315)(272,316)(273,289)(274,290)
(275,292)(276,291)(277,294)(278,293)(279,295)(280,296)(281,302)(282,301)
(283,303)(284,304)(285,298)(286,297)(287,299)(288,300);;
s1 := ( 1,145)( 2,148)( 3,147)( 4,146)( 5,157)( 6,160)( 7,159)( 8,158)
( 9,155)( 10,154)( 11,153)( 12,156)( 13,149)( 14,152)( 15,151)( 16,150)
( 17,177)( 18,180)( 19,179)( 20,178)( 21,189)( 22,192)( 23,191)( 24,190)
( 25,187)( 26,186)( 27,185)( 28,188)( 29,181)( 30,184)( 31,183)( 32,182)
( 33,161)( 34,164)( 35,163)( 36,162)( 37,173)( 38,176)( 39,175)( 40,174)
( 41,171)( 42,170)( 43,169)( 44,172)( 45,165)( 46,168)( 47,167)( 48,166)
( 49,273)( 50,276)( 51,275)( 52,274)( 53,285)( 54,288)( 55,287)( 56,286)
( 57,283)( 58,282)( 59,281)( 60,284)( 61,277)( 62,280)( 63,279)( 64,278)
( 65,257)( 66,260)( 67,259)( 68,258)( 69,269)( 70,272)( 71,271)( 72,270)
( 73,267)( 74,266)( 75,265)( 76,268)( 77,261)( 78,264)( 79,263)( 80,262)
( 81,241)( 82,244)( 83,243)( 84,242)( 85,253)( 86,256)( 87,255)( 88,254)
( 89,251)( 90,250)( 91,249)( 92,252)( 93,245)( 94,248)( 95,247)( 96,246)
( 97,225)( 98,228)( 99,227)(100,226)(101,237)(102,240)(103,239)(104,238)
(105,235)(106,234)(107,233)(108,236)(109,229)(110,232)(111,231)(112,230)
(113,209)(114,212)(115,211)(116,210)(117,221)(118,224)(119,223)(120,222)
(121,219)(122,218)(123,217)(124,220)(125,213)(126,216)(127,215)(128,214)
(129,193)(130,196)(131,195)(132,194)(133,205)(134,208)(135,207)(136,206)
(137,203)(138,202)(139,201)(140,204)(141,197)(142,200)(143,199)(144,198)
(289,417)(290,420)(291,419)(292,418)(293,429)(294,432)(295,431)(296,430)
(297,427)(298,426)(299,425)(300,428)(301,421)(302,424)(303,423)(304,422)
(305,401)(306,404)(307,403)(308,402)(309,413)(310,416)(311,415)(312,414)
(313,411)(314,410)(315,409)(316,412)(317,405)(318,408)(319,407)(320,406)
(321,385)(322,388)(323,387)(324,386)(325,397)(326,400)(327,399)(328,398)
(329,395)(330,394)(331,393)(332,396)(333,389)(334,392)(335,391)(336,390)
(337,369)(338,372)(339,371)(340,370)(341,381)(342,384)(343,383)(344,382)
(345,379)(346,378)(347,377)(348,380)(349,373)(350,376)(351,375)(352,374)
(354,356)(357,365)(358,368)(359,367)(360,366)(361,363);;
s2 := ( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9, 13)( 10, 14)( 11, 15)( 12, 16)
( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 29)( 26, 30)( 27, 31)( 28, 32)
( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 45)( 42, 46)( 43, 47)( 44, 48)
( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 61)( 58, 62)( 59, 63)( 60, 64)
( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 77)( 74, 78)( 75, 79)( 76, 80)
( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 93)( 90, 94)( 91, 95)( 92, 96)
( 97,103)( 98,104)( 99,101)(100,102)(105,109)(106,110)(107,111)(108,112)
(113,119)(114,120)(115,117)(116,118)(121,125)(122,126)(123,127)(124,128)
(129,135)(130,136)(131,133)(132,134)(137,141)(138,142)(139,143)(140,144)
(145,151)(146,152)(147,149)(148,150)(153,157)(154,158)(155,159)(156,160)
(161,167)(162,168)(163,165)(164,166)(169,173)(170,174)(171,175)(172,176)
(177,183)(178,184)(179,181)(180,182)(185,189)(186,190)(187,191)(188,192)
(193,199)(194,200)(195,197)(196,198)(201,205)(202,206)(203,207)(204,208)
(209,215)(210,216)(211,213)(212,214)(217,221)(218,222)(219,223)(220,224)
(225,231)(226,232)(227,229)(228,230)(233,237)(234,238)(235,239)(236,240)
(241,247)(242,248)(243,245)(244,246)(249,253)(250,254)(251,255)(252,256)
(257,263)(258,264)(259,261)(260,262)(265,269)(266,270)(267,271)(268,272)
(273,279)(274,280)(275,277)(276,278)(281,285)(282,286)(283,287)(284,288)
(289,295)(290,296)(291,293)(292,294)(297,301)(298,302)(299,303)(300,304)
(305,311)(306,312)(307,309)(308,310)(313,317)(314,318)(315,319)(316,320)
(321,327)(322,328)(323,325)(324,326)(329,333)(330,334)(331,335)(332,336)
(337,343)(338,344)(339,341)(340,342)(345,349)(346,350)(347,351)(348,352)
(353,359)(354,360)(355,357)(356,358)(361,365)(362,366)(363,367)(364,368)
(369,375)(370,376)(371,373)(372,374)(377,381)(378,382)(379,383)(380,384)
(385,391)(386,392)(387,389)(388,390)(393,397)(394,398)(395,399)(396,400)
(401,407)(402,408)(403,405)(404,406)(409,413)(410,414)(411,415)(412,416)
(417,423)(418,424)(419,421)(420,422)(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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*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*s2*s1*s0*s2*s1*s0*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 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 5, 6)( 9, 14)( 10, 13)( 11, 15)( 12, 16)( 17, 33)
( 18, 34)( 19, 36)( 20, 35)( 21, 38)( 22, 37)( 23, 39)( 24, 40)( 25, 46)
( 26, 45)( 27, 47)( 28, 48)( 29, 42)( 30, 41)( 31, 43)( 32, 44)( 49,129)
( 50,130)( 51,132)( 52,131)( 53,134)( 54,133)( 55,135)( 56,136)( 57,142)
( 58,141)( 59,143)( 60,144)( 61,138)( 62,137)( 63,139)( 64,140)( 65,113)
( 66,114)( 67,116)( 68,115)( 69,118)( 70,117)( 71,119)( 72,120)( 73,126)
( 74,125)( 75,127)( 76,128)( 77,122)( 78,121)( 79,123)( 80,124)( 81, 97)
( 82, 98)( 83,100)( 84, 99)( 85,102)( 86,101)( 87,103)( 88,104)( 89,110)
( 90,109)( 91,111)( 92,112)( 93,106)( 94,105)( 95,107)( 96,108)(145,417)
(146,418)(147,420)(148,419)(149,422)(150,421)(151,423)(152,424)(153,430)
(154,429)(155,431)(156,432)(157,426)(158,425)(159,427)(160,428)(161,401)
(162,402)(163,404)(164,403)(165,406)(166,405)(167,407)(168,408)(169,414)
(170,413)(171,415)(172,416)(173,410)(174,409)(175,411)(176,412)(177,385)
(178,386)(179,388)(180,387)(181,390)(182,389)(183,391)(184,392)(185,398)
(186,397)(187,399)(188,400)(189,394)(190,393)(191,395)(192,396)(193,369)
(194,370)(195,372)(196,371)(197,374)(198,373)(199,375)(200,376)(201,382)
(202,381)(203,383)(204,384)(205,378)(206,377)(207,379)(208,380)(209,353)
(210,354)(211,356)(212,355)(213,358)(214,357)(215,359)(216,360)(217,366)
(218,365)(219,367)(220,368)(221,362)(222,361)(223,363)(224,364)(225,337)
(226,338)(227,340)(228,339)(229,342)(230,341)(231,343)(232,344)(233,350)
(234,349)(235,351)(236,352)(237,346)(238,345)(239,347)(240,348)(241,321)
(242,322)(243,324)(244,323)(245,326)(246,325)(247,327)(248,328)(249,334)
(250,333)(251,335)(252,336)(253,330)(254,329)(255,331)(256,332)(257,305)
(258,306)(259,308)(260,307)(261,310)(262,309)(263,311)(264,312)(265,318)
(266,317)(267,319)(268,320)(269,314)(270,313)(271,315)(272,316)(273,289)
(274,290)(275,292)(276,291)(277,294)(278,293)(279,295)(280,296)(281,302)
(282,301)(283,303)(284,304)(285,298)(286,297)(287,299)(288,300);
s1 := Sym(432)!( 1,145)( 2,148)( 3,147)( 4,146)( 5,157)( 6,160)( 7,159)
( 8,158)( 9,155)( 10,154)( 11,153)( 12,156)( 13,149)( 14,152)( 15,151)
( 16,150)( 17,177)( 18,180)( 19,179)( 20,178)( 21,189)( 22,192)( 23,191)
( 24,190)( 25,187)( 26,186)( 27,185)( 28,188)( 29,181)( 30,184)( 31,183)
( 32,182)( 33,161)( 34,164)( 35,163)( 36,162)( 37,173)( 38,176)( 39,175)
( 40,174)( 41,171)( 42,170)( 43,169)( 44,172)( 45,165)( 46,168)( 47,167)
( 48,166)( 49,273)( 50,276)( 51,275)( 52,274)( 53,285)( 54,288)( 55,287)
( 56,286)( 57,283)( 58,282)( 59,281)( 60,284)( 61,277)( 62,280)( 63,279)
( 64,278)( 65,257)( 66,260)( 67,259)( 68,258)( 69,269)( 70,272)( 71,271)
( 72,270)( 73,267)( 74,266)( 75,265)( 76,268)( 77,261)( 78,264)( 79,263)
( 80,262)( 81,241)( 82,244)( 83,243)( 84,242)( 85,253)( 86,256)( 87,255)
( 88,254)( 89,251)( 90,250)( 91,249)( 92,252)( 93,245)( 94,248)( 95,247)
( 96,246)( 97,225)( 98,228)( 99,227)(100,226)(101,237)(102,240)(103,239)
(104,238)(105,235)(106,234)(107,233)(108,236)(109,229)(110,232)(111,231)
(112,230)(113,209)(114,212)(115,211)(116,210)(117,221)(118,224)(119,223)
(120,222)(121,219)(122,218)(123,217)(124,220)(125,213)(126,216)(127,215)
(128,214)(129,193)(130,196)(131,195)(132,194)(133,205)(134,208)(135,207)
(136,206)(137,203)(138,202)(139,201)(140,204)(141,197)(142,200)(143,199)
(144,198)(289,417)(290,420)(291,419)(292,418)(293,429)(294,432)(295,431)
(296,430)(297,427)(298,426)(299,425)(300,428)(301,421)(302,424)(303,423)
(304,422)(305,401)(306,404)(307,403)(308,402)(309,413)(310,416)(311,415)
(312,414)(313,411)(314,410)(315,409)(316,412)(317,405)(318,408)(319,407)
(320,406)(321,385)(322,388)(323,387)(324,386)(325,397)(326,400)(327,399)
(328,398)(329,395)(330,394)(331,393)(332,396)(333,389)(334,392)(335,391)
(336,390)(337,369)(338,372)(339,371)(340,370)(341,381)(342,384)(343,383)
(344,382)(345,379)(346,378)(347,377)(348,380)(349,373)(350,376)(351,375)
(352,374)(354,356)(357,365)(358,368)(359,367)(360,366)(361,363);
s2 := Sym(432)!( 1, 7)( 2, 8)( 3, 5)( 4, 6)( 9, 13)( 10, 14)( 11, 15)
( 12, 16)( 17, 23)( 18, 24)( 19, 21)( 20, 22)( 25, 29)( 26, 30)( 27, 31)
( 28, 32)( 33, 39)( 34, 40)( 35, 37)( 36, 38)( 41, 45)( 42, 46)( 43, 47)
( 44, 48)( 49, 55)( 50, 56)( 51, 53)( 52, 54)( 57, 61)( 58, 62)( 59, 63)
( 60, 64)( 65, 71)( 66, 72)( 67, 69)( 68, 70)( 73, 77)( 74, 78)( 75, 79)
( 76, 80)( 81, 87)( 82, 88)( 83, 85)( 84, 86)( 89, 93)( 90, 94)( 91, 95)
( 92, 96)( 97,103)( 98,104)( 99,101)(100,102)(105,109)(106,110)(107,111)
(108,112)(113,119)(114,120)(115,117)(116,118)(121,125)(122,126)(123,127)
(124,128)(129,135)(130,136)(131,133)(132,134)(137,141)(138,142)(139,143)
(140,144)(145,151)(146,152)(147,149)(148,150)(153,157)(154,158)(155,159)
(156,160)(161,167)(162,168)(163,165)(164,166)(169,173)(170,174)(171,175)
(172,176)(177,183)(178,184)(179,181)(180,182)(185,189)(186,190)(187,191)
(188,192)(193,199)(194,200)(195,197)(196,198)(201,205)(202,206)(203,207)
(204,208)(209,215)(210,216)(211,213)(212,214)(217,221)(218,222)(219,223)
(220,224)(225,231)(226,232)(227,229)(228,230)(233,237)(234,238)(235,239)
(236,240)(241,247)(242,248)(243,245)(244,246)(249,253)(250,254)(251,255)
(252,256)(257,263)(258,264)(259,261)(260,262)(265,269)(266,270)(267,271)
(268,272)(273,279)(274,280)(275,277)(276,278)(281,285)(282,286)(283,287)
(284,288)(289,295)(290,296)(291,293)(292,294)(297,301)(298,302)(299,303)
(300,304)(305,311)(306,312)(307,309)(308,310)(313,317)(314,318)(315,319)
(316,320)(321,327)(322,328)(323,325)(324,326)(329,333)(330,334)(331,335)
(332,336)(337,343)(338,344)(339,341)(340,342)(345,349)(346,350)(347,351)
(348,352)(353,359)(354,360)(355,357)(356,358)(361,365)(362,366)(363,367)
(364,368)(369,375)(370,376)(371,373)(372,374)(377,381)(378,382)(379,383)
(380,384)(385,391)(386,392)(387,389)(388,390)(393,397)(394,398)(395,399)
(396,400)(401,407)(402,408)(403,405)(404,406)(409,413)(410,414)(411,415)
(412,416)(417,423)(418,424)(419,421)(420,422)(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, s2*s0*s1*s2*s1*s2*s0*s1*s2*s0*s1*s2*s1*s2*s0*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2,
s0*s1*s2*s1*s2*s0*s1*s0*s1*s2*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*s2*s1*s0*s2*s1*s0*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 >;
References : None.
to this polytope