Polytope of Type {8,54}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {8,54}*1728a
if this polytope has a name.
Group : SmallGroup(1728,2198)
Rank : 3
Schlafli Type : {8,54}
Number of vertices, edges, etc : 16, 432, 108
Order of s0s1s2 : 27
Order of s0s1s2s1 : 8
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 : {8,18}*576a
4-fold quotients : {4,54}*432c
8-fold quotients : {4,27}*216
9-fold quotients : {8,6}*192a
12-fold quotients : {4,18}*144c
24-fold quotients : {4,9}*72
36-fold quotients : {4,6}*48b
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, 14)( 6, 13)( 7, 16)( 8, 15)
( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 30)( 22, 29)( 23, 32)( 24, 31)
( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 46)( 38, 45)( 39, 48)( 40, 47)
( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 62)( 54, 61)( 55, 64)( 56, 63)
( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 78)( 70, 77)( 71, 80)( 72, 79)
( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 87, 96)( 88, 95)
( 97,105)( 98,106)( 99,107)(100,108)(101,110)(102,109)(103,112)(104,111)
(113,121)(114,122)(115,123)(116,124)(117,126)(118,125)(119,128)(120,127)
(129,137)(130,138)(131,139)(132,140)(133,142)(134,141)(135,144)(136,143)
(145,153)(146,154)(147,155)(148,156)(149,158)(150,157)(151,160)(152,159)
(161,169)(162,170)(163,171)(164,172)(165,174)(166,173)(167,176)(168,175)
(177,185)(178,186)(179,187)(180,188)(181,190)(182,189)(183,192)(184,191)
(193,201)(194,202)(195,203)(196,204)(197,206)(198,205)(199,208)(200,207)
(209,217)(210,218)(211,219)(212,220)(213,222)(214,221)(215,224)(216,223)
(225,233)(226,234)(227,235)(228,236)(229,238)(230,237)(231,240)(232,239)
(241,249)(242,250)(243,251)(244,252)(245,254)(246,253)(247,256)(248,255)
(257,265)(258,266)(259,267)(260,268)(261,270)(262,269)(263,272)(264,271)
(273,281)(274,282)(275,283)(276,284)(277,286)(278,285)(279,288)(280,287)
(289,297)(290,298)(291,299)(292,300)(293,302)(294,301)(295,304)(296,303)
(305,313)(306,314)(307,315)(308,316)(309,318)(310,317)(311,320)(312,319)
(321,329)(322,330)(323,331)(324,332)(325,334)(326,333)(327,336)(328,335)
(337,345)(338,346)(339,347)(340,348)(341,350)(342,349)(343,352)(344,351)
(353,361)(354,362)(355,363)(356,364)(357,366)(358,365)(359,368)(360,367)
(369,377)(370,378)(371,379)(372,380)(373,382)(374,381)(375,384)(376,383)
(385,393)(386,394)(387,395)(388,396)(389,398)(390,397)(391,400)(392,399)
(401,409)(402,410)(403,411)(404,412)(405,414)(406,413)(407,416)(408,415)
(417,425)(418,426)(419,427)(420,428)(421,430)(422,429)(423,432)(424,431);;
s1 := ( 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);;
s2 := ( 1,153)( 2,156)( 3,155)( 4,154)( 5,151)( 6,150)( 7,149)( 8,152)
( 9,145)( 10,148)( 11,147)( 12,146)( 13,157)( 14,160)( 15,159)( 16,158)
( 17,185)( 18,188)( 19,187)( 20,186)( 21,183)( 22,182)( 23,181)( 24,184)
( 25,177)( 26,180)( 27,179)( 28,178)( 29,189)( 30,192)( 31,191)( 32,190)
( 33,169)( 34,172)( 35,171)( 36,170)( 37,167)( 38,166)( 39,165)( 40,168)
( 41,161)( 42,164)( 43,163)( 44,162)( 45,173)( 46,176)( 47,175)( 48,174)
( 49,281)( 50,284)( 51,283)( 52,282)( 53,279)( 54,278)( 55,277)( 56,280)
( 57,273)( 58,276)( 59,275)( 60,274)( 61,285)( 62,288)( 63,287)( 64,286)
( 65,265)( 66,268)( 67,267)( 68,266)( 69,263)( 70,262)( 71,261)( 72,264)
( 73,257)( 74,260)( 75,259)( 76,258)( 77,269)( 78,272)( 79,271)( 80,270)
( 81,249)( 82,252)( 83,251)( 84,250)( 85,247)( 86,246)( 87,245)( 88,248)
( 89,241)( 90,244)( 91,243)( 92,242)( 93,253)( 94,256)( 95,255)( 96,254)
( 97,233)( 98,236)( 99,235)(100,234)(101,231)(102,230)(103,229)(104,232)
(105,225)(106,228)(107,227)(108,226)(109,237)(110,240)(111,239)(112,238)
(113,217)(114,220)(115,219)(116,218)(117,215)(118,214)(119,213)(120,216)
(121,209)(122,212)(123,211)(124,210)(125,221)(126,224)(127,223)(128,222)
(129,201)(130,204)(131,203)(132,202)(133,199)(134,198)(135,197)(136,200)
(137,193)(138,196)(139,195)(140,194)(141,205)(142,208)(143,207)(144,206)
(289,425)(290,428)(291,427)(292,426)(293,423)(294,422)(295,421)(296,424)
(297,417)(298,420)(299,419)(300,418)(301,429)(302,432)(303,431)(304,430)
(305,409)(306,412)(307,411)(308,410)(309,407)(310,406)(311,405)(312,408)
(313,401)(314,404)(315,403)(316,402)(317,413)(318,416)(319,415)(320,414)
(321,393)(322,396)(323,395)(324,394)(325,391)(326,390)(327,389)(328,392)
(329,385)(330,388)(331,387)(332,386)(333,397)(334,400)(335,399)(336,398)
(337,377)(338,380)(339,379)(340,378)(341,375)(342,374)(343,373)(344,376)
(345,369)(346,372)(347,371)(348,370)(349,381)(350,384)(351,383)(352,382)
(353,361)(354,364)(355,363)(356,362)(357,359)(366,368);;
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*s0*s1*s0*s1*s2*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*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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, 14)( 6, 13)( 7, 16)
( 8, 15)( 17, 25)( 18, 26)( 19, 27)( 20, 28)( 21, 30)( 22, 29)( 23, 32)
( 24, 31)( 33, 41)( 34, 42)( 35, 43)( 36, 44)( 37, 46)( 38, 45)( 39, 48)
( 40, 47)( 49, 57)( 50, 58)( 51, 59)( 52, 60)( 53, 62)( 54, 61)( 55, 64)
( 56, 63)( 65, 73)( 66, 74)( 67, 75)( 68, 76)( 69, 78)( 70, 77)( 71, 80)
( 72, 79)( 81, 89)( 82, 90)( 83, 91)( 84, 92)( 85, 94)( 86, 93)( 87, 96)
( 88, 95)( 97,105)( 98,106)( 99,107)(100,108)(101,110)(102,109)(103,112)
(104,111)(113,121)(114,122)(115,123)(116,124)(117,126)(118,125)(119,128)
(120,127)(129,137)(130,138)(131,139)(132,140)(133,142)(134,141)(135,144)
(136,143)(145,153)(146,154)(147,155)(148,156)(149,158)(150,157)(151,160)
(152,159)(161,169)(162,170)(163,171)(164,172)(165,174)(166,173)(167,176)
(168,175)(177,185)(178,186)(179,187)(180,188)(181,190)(182,189)(183,192)
(184,191)(193,201)(194,202)(195,203)(196,204)(197,206)(198,205)(199,208)
(200,207)(209,217)(210,218)(211,219)(212,220)(213,222)(214,221)(215,224)
(216,223)(225,233)(226,234)(227,235)(228,236)(229,238)(230,237)(231,240)
(232,239)(241,249)(242,250)(243,251)(244,252)(245,254)(246,253)(247,256)
(248,255)(257,265)(258,266)(259,267)(260,268)(261,270)(262,269)(263,272)
(264,271)(273,281)(274,282)(275,283)(276,284)(277,286)(278,285)(279,288)
(280,287)(289,297)(290,298)(291,299)(292,300)(293,302)(294,301)(295,304)
(296,303)(305,313)(306,314)(307,315)(308,316)(309,318)(310,317)(311,320)
(312,319)(321,329)(322,330)(323,331)(324,332)(325,334)(326,333)(327,336)
(328,335)(337,345)(338,346)(339,347)(340,348)(341,350)(342,349)(343,352)
(344,351)(353,361)(354,362)(355,363)(356,364)(357,366)(358,365)(359,368)
(360,367)(369,377)(370,378)(371,379)(372,380)(373,382)(374,381)(375,384)
(376,383)(385,393)(386,394)(387,395)(388,396)(389,398)(390,397)(391,400)
(392,399)(401,409)(402,410)(403,411)(404,412)(405,414)(406,413)(407,416)
(408,415)(417,425)(418,426)(419,427)(420,428)(421,430)(422,429)(423,432)
(424,431);
s1 := 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);
s2 := Sym(432)!( 1,153)( 2,156)( 3,155)( 4,154)( 5,151)( 6,150)( 7,149)
( 8,152)( 9,145)( 10,148)( 11,147)( 12,146)( 13,157)( 14,160)( 15,159)
( 16,158)( 17,185)( 18,188)( 19,187)( 20,186)( 21,183)( 22,182)( 23,181)
( 24,184)( 25,177)( 26,180)( 27,179)( 28,178)( 29,189)( 30,192)( 31,191)
( 32,190)( 33,169)( 34,172)( 35,171)( 36,170)( 37,167)( 38,166)( 39,165)
( 40,168)( 41,161)( 42,164)( 43,163)( 44,162)( 45,173)( 46,176)( 47,175)
( 48,174)( 49,281)( 50,284)( 51,283)( 52,282)( 53,279)( 54,278)( 55,277)
( 56,280)( 57,273)( 58,276)( 59,275)( 60,274)( 61,285)( 62,288)( 63,287)
( 64,286)( 65,265)( 66,268)( 67,267)( 68,266)( 69,263)( 70,262)( 71,261)
( 72,264)( 73,257)( 74,260)( 75,259)( 76,258)( 77,269)( 78,272)( 79,271)
( 80,270)( 81,249)( 82,252)( 83,251)( 84,250)( 85,247)( 86,246)( 87,245)
( 88,248)( 89,241)( 90,244)( 91,243)( 92,242)( 93,253)( 94,256)( 95,255)
( 96,254)( 97,233)( 98,236)( 99,235)(100,234)(101,231)(102,230)(103,229)
(104,232)(105,225)(106,228)(107,227)(108,226)(109,237)(110,240)(111,239)
(112,238)(113,217)(114,220)(115,219)(116,218)(117,215)(118,214)(119,213)
(120,216)(121,209)(122,212)(123,211)(124,210)(125,221)(126,224)(127,223)
(128,222)(129,201)(130,204)(131,203)(132,202)(133,199)(134,198)(135,197)
(136,200)(137,193)(138,196)(139,195)(140,194)(141,205)(142,208)(143,207)
(144,206)(289,425)(290,428)(291,427)(292,426)(293,423)(294,422)(295,421)
(296,424)(297,417)(298,420)(299,419)(300,418)(301,429)(302,432)(303,431)
(304,430)(305,409)(306,412)(307,411)(308,410)(309,407)(310,406)(311,405)
(312,408)(313,401)(314,404)(315,403)(316,402)(317,413)(318,416)(319,415)
(320,414)(321,393)(322,396)(323,395)(324,394)(325,391)(326,390)(327,389)
(328,392)(329,385)(330,388)(331,387)(332,386)(333,397)(334,400)(335,399)
(336,398)(337,377)(338,380)(339,379)(340,378)(341,375)(342,374)(343,373)
(344,376)(345,369)(346,372)(347,371)(348,370)(349,381)(350,384)(351,383)
(352,382)(353,361)(354,364)(355,363)(356,362)(357,359)(366,368);
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*s0*s1*s0*s1*s2*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*s2*s0*s1*s0*s1*s2*s1*s2*s1*s0*s1*s0*s2*s1,
s1*s2*s1*s2*s1*s2*s1*s2*s1*s0*s2*s1*s0*s2*s1*s0*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