Polytope of Type {54,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,4}*864
if this polytope has a name.
Group : SmallGroup(864,1916)
Rank : 3
Schlafli Type : {54,4}
Number of vertices, edges, etc : 108, 216, 8
Order of s0s1s2 : 54
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
Self-Petrie
Related Polytopes :
Facet
Vertex Figure
Dual
Petrial
Skewing Operation
Facet Of :
{54,4,2} of size 1728
Vertex Figure Of :
{2,54,4} of size 1728
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {27,4}*432, {54,4}*432b, {54,4}*432c
3-fold quotients : {18,4}*288
4-fold quotients : {27,4}*216, {54,2}*216
6-fold quotients : {9,4}*144, {18,4}*144b, {18,4}*144c
8-fold quotients : {27,2}*108
9-fold quotients : {6,4}*96
12-fold quotients : {9,4}*72, {18,2}*72
18-fold quotients : {3,4}*48, {6,4}*48b, {6,4}*48c
24-fold quotients : {9,2}*36
36-fold quotients : {3,4}*24, {6,2}*24
72-fold quotients : {3,2}*12
108-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
2-fold covers : {108,4}*1728b, {54,4}*1728b, {108,4}*1728c, {54,8}*1728b, {54,8}*1728c
Permutation Representation (GAP) :
s0 := ( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 29)( 14, 31)( 15, 30)
( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)( 23, 34)
( 24, 36)( 37, 97)( 38, 99)( 39, 98)( 40,100)( 41,105)( 42,107)( 43,106)
( 44,108)( 45,101)( 46,103)( 47,102)( 48,104)( 49, 85)( 50, 87)( 51, 86)
( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)( 59, 90)
( 60, 92)( 61, 73)( 62, 75)( 63, 74)( 64, 76)( 65, 81)( 66, 83)( 67, 82)
( 68, 84)( 69, 77)( 70, 79)( 71, 78)( 72, 80)(110,111)(113,117)(114,119)
(115,118)(116,120)(121,137)(122,139)(123,138)(124,140)(125,133)(126,135)
(127,134)(128,136)(129,141)(130,143)(131,142)(132,144)(145,205)(146,207)
(147,206)(148,208)(149,213)(150,215)(151,214)(152,216)(153,209)(154,211)
(155,210)(156,212)(157,193)(158,195)(159,194)(160,196)(161,201)(162,203)
(163,202)(164,204)(165,197)(166,199)(167,198)(168,200)(169,181)(170,183)
(171,182)(172,184)(173,189)(174,191)(175,190)(176,192)(177,185)(178,187)
(179,186)(180,188)(218,219)(221,225)(222,227)(223,226)(224,228)(229,245)
(230,247)(231,246)(232,248)(233,241)(234,243)(235,242)(236,244)(237,249)
(238,251)(239,250)(240,252)(253,313)(254,315)(255,314)(256,316)(257,321)
(258,323)(259,322)(260,324)(261,317)(262,319)(263,318)(264,320)(265,301)
(266,303)(267,302)(268,304)(269,309)(270,311)(271,310)(272,312)(273,305)
(274,307)(275,306)(276,308)(277,289)(278,291)(279,290)(280,292)(281,297)
(282,299)(283,298)(284,300)(285,293)(286,295)(287,294)(288,296)(326,327)
(329,333)(330,335)(331,334)(332,336)(337,353)(338,355)(339,354)(340,356)
(341,349)(342,351)(343,350)(344,352)(345,357)(346,359)(347,358)(348,360)
(361,421)(362,423)(363,422)(364,424)(365,429)(366,431)(367,430)(368,432)
(369,425)(370,427)(371,426)(372,428)(373,409)(374,411)(375,410)(376,412)
(377,417)(378,419)(379,418)(380,420)(381,413)(382,415)(383,414)(384,416)
(385,397)(386,399)(387,398)(388,400)(389,405)(390,407)(391,406)(392,408)
(393,401)(394,403)(395,402)(396,404);;
s1 := ( 1,253)( 2,254)( 3,256)( 4,255)( 5,261)( 6,262)( 7,264)( 8,263)
( 9,257)( 10,258)( 11,260)( 12,259)( 13,281)( 14,282)( 15,284)( 16,283)
( 17,277)( 18,278)( 19,280)( 20,279)( 21,285)( 22,286)( 23,288)( 24,287)
( 25,269)( 26,270)( 27,272)( 28,271)( 29,265)( 30,266)( 31,268)( 32,267)
( 33,273)( 34,274)( 35,276)( 36,275)( 37,217)( 38,218)( 39,220)( 40,219)
( 41,225)( 42,226)( 43,228)( 44,227)( 45,221)( 46,222)( 47,224)( 48,223)
( 49,245)( 50,246)( 51,248)( 52,247)( 53,241)( 54,242)( 55,244)( 56,243)
( 57,249)( 58,250)( 59,252)( 60,251)( 61,233)( 62,234)( 63,236)( 64,235)
( 65,229)( 66,230)( 67,232)( 68,231)( 69,237)( 70,238)( 71,240)( 72,239)
( 73,313)( 74,314)( 75,316)( 76,315)( 77,321)( 78,322)( 79,324)( 80,323)
( 81,317)( 82,318)( 83,320)( 84,319)( 85,301)( 86,302)( 87,304)( 88,303)
( 89,309)( 90,310)( 91,312)( 92,311)( 93,305)( 94,306)( 95,308)( 96,307)
( 97,289)( 98,290)( 99,292)(100,291)(101,297)(102,298)(103,300)(104,299)
(105,293)(106,294)(107,296)(108,295)(109,361)(110,362)(111,364)(112,363)
(113,369)(114,370)(115,372)(116,371)(117,365)(118,366)(119,368)(120,367)
(121,389)(122,390)(123,392)(124,391)(125,385)(126,386)(127,388)(128,387)
(129,393)(130,394)(131,396)(132,395)(133,377)(134,378)(135,380)(136,379)
(137,373)(138,374)(139,376)(140,375)(141,381)(142,382)(143,384)(144,383)
(145,325)(146,326)(147,328)(148,327)(149,333)(150,334)(151,336)(152,335)
(153,329)(154,330)(155,332)(156,331)(157,353)(158,354)(159,356)(160,355)
(161,349)(162,350)(163,352)(164,351)(165,357)(166,358)(167,360)(168,359)
(169,341)(170,342)(171,344)(172,343)(173,337)(174,338)(175,340)(176,339)
(177,345)(178,346)(179,348)(180,347)(181,421)(182,422)(183,424)(184,423)
(185,429)(186,430)(187,432)(188,431)(189,425)(190,426)(191,428)(192,427)
(193,409)(194,410)(195,412)(196,411)(197,417)(198,418)(199,420)(200,419)
(201,413)(202,414)(203,416)(204,415)(205,397)(206,398)(207,400)(208,399)
(209,405)(210,406)(211,408)(212,407)(213,401)(214,402)(215,404)(216,403);;
s2 := ( 1,112)( 2,111)( 3,110)( 4,109)( 5,116)( 6,115)( 7,114)( 8,113)
( 9,120)( 10,119)( 11,118)( 12,117)( 13,124)( 14,123)( 15,122)( 16,121)
( 17,128)( 18,127)( 19,126)( 20,125)( 21,132)( 22,131)( 23,130)( 24,129)
( 25,136)( 26,135)( 27,134)( 28,133)( 29,140)( 30,139)( 31,138)( 32,137)
( 33,144)( 34,143)( 35,142)( 36,141)( 37,148)( 38,147)( 39,146)( 40,145)
( 41,152)( 42,151)( 43,150)( 44,149)( 45,156)( 46,155)( 47,154)( 48,153)
( 49,160)( 50,159)( 51,158)( 52,157)( 53,164)( 54,163)( 55,162)( 56,161)
( 57,168)( 58,167)( 59,166)( 60,165)( 61,172)( 62,171)( 63,170)( 64,169)
( 65,176)( 66,175)( 67,174)( 68,173)( 69,180)( 70,179)( 71,178)( 72,177)
( 73,184)( 74,183)( 75,182)( 76,181)( 77,188)( 78,187)( 79,186)( 80,185)
( 81,192)( 82,191)( 83,190)( 84,189)( 85,196)( 86,195)( 87,194)( 88,193)
( 89,200)( 90,199)( 91,198)( 92,197)( 93,204)( 94,203)( 95,202)( 96,201)
( 97,208)( 98,207)( 99,206)(100,205)(101,212)(102,211)(103,210)(104,209)
(105,216)(106,215)(107,214)(108,213)(217,328)(218,327)(219,326)(220,325)
(221,332)(222,331)(223,330)(224,329)(225,336)(226,335)(227,334)(228,333)
(229,340)(230,339)(231,338)(232,337)(233,344)(234,343)(235,342)(236,341)
(237,348)(238,347)(239,346)(240,345)(241,352)(242,351)(243,350)(244,349)
(245,356)(246,355)(247,354)(248,353)(249,360)(250,359)(251,358)(252,357)
(253,364)(254,363)(255,362)(256,361)(257,368)(258,367)(259,366)(260,365)
(261,372)(262,371)(263,370)(264,369)(265,376)(266,375)(267,374)(268,373)
(269,380)(270,379)(271,378)(272,377)(273,384)(274,383)(275,382)(276,381)
(277,388)(278,387)(279,386)(280,385)(281,392)(282,391)(283,390)(284,389)
(285,396)(286,395)(287,394)(288,393)(289,400)(290,399)(291,398)(292,397)
(293,404)(294,403)(295,402)(296,401)(297,408)(298,407)(299,406)(300,405)
(301,412)(302,411)(303,410)(304,409)(305,416)(306,415)(307,414)(308,413)
(309,420)(310,419)(311,418)(312,417)(313,424)(314,423)(315,422)(316,421)
(317,428)(318,427)(319,426)(320,425)(321,432)(322,431)(323,430)(324,429);;
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*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*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)!( 2, 3)( 5, 9)( 6, 11)( 7, 10)( 8, 12)( 13, 29)( 14, 31)
( 15, 30)( 16, 32)( 17, 25)( 18, 27)( 19, 26)( 20, 28)( 21, 33)( 22, 35)
( 23, 34)( 24, 36)( 37, 97)( 38, 99)( 39, 98)( 40,100)( 41,105)( 42,107)
( 43,106)( 44,108)( 45,101)( 46,103)( 47,102)( 48,104)( 49, 85)( 50, 87)
( 51, 86)( 52, 88)( 53, 93)( 54, 95)( 55, 94)( 56, 96)( 57, 89)( 58, 91)
( 59, 90)( 60, 92)( 61, 73)( 62, 75)( 63, 74)( 64, 76)( 65, 81)( 66, 83)
( 67, 82)( 68, 84)( 69, 77)( 70, 79)( 71, 78)( 72, 80)(110,111)(113,117)
(114,119)(115,118)(116,120)(121,137)(122,139)(123,138)(124,140)(125,133)
(126,135)(127,134)(128,136)(129,141)(130,143)(131,142)(132,144)(145,205)
(146,207)(147,206)(148,208)(149,213)(150,215)(151,214)(152,216)(153,209)
(154,211)(155,210)(156,212)(157,193)(158,195)(159,194)(160,196)(161,201)
(162,203)(163,202)(164,204)(165,197)(166,199)(167,198)(168,200)(169,181)
(170,183)(171,182)(172,184)(173,189)(174,191)(175,190)(176,192)(177,185)
(178,187)(179,186)(180,188)(218,219)(221,225)(222,227)(223,226)(224,228)
(229,245)(230,247)(231,246)(232,248)(233,241)(234,243)(235,242)(236,244)
(237,249)(238,251)(239,250)(240,252)(253,313)(254,315)(255,314)(256,316)
(257,321)(258,323)(259,322)(260,324)(261,317)(262,319)(263,318)(264,320)
(265,301)(266,303)(267,302)(268,304)(269,309)(270,311)(271,310)(272,312)
(273,305)(274,307)(275,306)(276,308)(277,289)(278,291)(279,290)(280,292)
(281,297)(282,299)(283,298)(284,300)(285,293)(286,295)(287,294)(288,296)
(326,327)(329,333)(330,335)(331,334)(332,336)(337,353)(338,355)(339,354)
(340,356)(341,349)(342,351)(343,350)(344,352)(345,357)(346,359)(347,358)
(348,360)(361,421)(362,423)(363,422)(364,424)(365,429)(366,431)(367,430)
(368,432)(369,425)(370,427)(371,426)(372,428)(373,409)(374,411)(375,410)
(376,412)(377,417)(378,419)(379,418)(380,420)(381,413)(382,415)(383,414)
(384,416)(385,397)(386,399)(387,398)(388,400)(389,405)(390,407)(391,406)
(392,408)(393,401)(394,403)(395,402)(396,404);
s1 := Sym(432)!( 1,253)( 2,254)( 3,256)( 4,255)( 5,261)( 6,262)( 7,264)
( 8,263)( 9,257)( 10,258)( 11,260)( 12,259)( 13,281)( 14,282)( 15,284)
( 16,283)( 17,277)( 18,278)( 19,280)( 20,279)( 21,285)( 22,286)( 23,288)
( 24,287)( 25,269)( 26,270)( 27,272)( 28,271)( 29,265)( 30,266)( 31,268)
( 32,267)( 33,273)( 34,274)( 35,276)( 36,275)( 37,217)( 38,218)( 39,220)
( 40,219)( 41,225)( 42,226)( 43,228)( 44,227)( 45,221)( 46,222)( 47,224)
( 48,223)( 49,245)( 50,246)( 51,248)( 52,247)( 53,241)( 54,242)( 55,244)
( 56,243)( 57,249)( 58,250)( 59,252)( 60,251)( 61,233)( 62,234)( 63,236)
( 64,235)( 65,229)( 66,230)( 67,232)( 68,231)( 69,237)( 70,238)( 71,240)
( 72,239)( 73,313)( 74,314)( 75,316)( 76,315)( 77,321)( 78,322)( 79,324)
( 80,323)( 81,317)( 82,318)( 83,320)( 84,319)( 85,301)( 86,302)( 87,304)
( 88,303)( 89,309)( 90,310)( 91,312)( 92,311)( 93,305)( 94,306)( 95,308)
( 96,307)( 97,289)( 98,290)( 99,292)(100,291)(101,297)(102,298)(103,300)
(104,299)(105,293)(106,294)(107,296)(108,295)(109,361)(110,362)(111,364)
(112,363)(113,369)(114,370)(115,372)(116,371)(117,365)(118,366)(119,368)
(120,367)(121,389)(122,390)(123,392)(124,391)(125,385)(126,386)(127,388)
(128,387)(129,393)(130,394)(131,396)(132,395)(133,377)(134,378)(135,380)
(136,379)(137,373)(138,374)(139,376)(140,375)(141,381)(142,382)(143,384)
(144,383)(145,325)(146,326)(147,328)(148,327)(149,333)(150,334)(151,336)
(152,335)(153,329)(154,330)(155,332)(156,331)(157,353)(158,354)(159,356)
(160,355)(161,349)(162,350)(163,352)(164,351)(165,357)(166,358)(167,360)
(168,359)(169,341)(170,342)(171,344)(172,343)(173,337)(174,338)(175,340)
(176,339)(177,345)(178,346)(179,348)(180,347)(181,421)(182,422)(183,424)
(184,423)(185,429)(186,430)(187,432)(188,431)(189,425)(190,426)(191,428)
(192,427)(193,409)(194,410)(195,412)(196,411)(197,417)(198,418)(199,420)
(200,419)(201,413)(202,414)(203,416)(204,415)(205,397)(206,398)(207,400)
(208,399)(209,405)(210,406)(211,408)(212,407)(213,401)(214,402)(215,404)
(216,403);
s2 := Sym(432)!( 1,112)( 2,111)( 3,110)( 4,109)( 5,116)( 6,115)( 7,114)
( 8,113)( 9,120)( 10,119)( 11,118)( 12,117)( 13,124)( 14,123)( 15,122)
( 16,121)( 17,128)( 18,127)( 19,126)( 20,125)( 21,132)( 22,131)( 23,130)
( 24,129)( 25,136)( 26,135)( 27,134)( 28,133)( 29,140)( 30,139)( 31,138)
( 32,137)( 33,144)( 34,143)( 35,142)( 36,141)( 37,148)( 38,147)( 39,146)
( 40,145)( 41,152)( 42,151)( 43,150)( 44,149)( 45,156)( 46,155)( 47,154)
( 48,153)( 49,160)( 50,159)( 51,158)( 52,157)( 53,164)( 54,163)( 55,162)
( 56,161)( 57,168)( 58,167)( 59,166)( 60,165)( 61,172)( 62,171)( 63,170)
( 64,169)( 65,176)( 66,175)( 67,174)( 68,173)( 69,180)( 70,179)( 71,178)
( 72,177)( 73,184)( 74,183)( 75,182)( 76,181)( 77,188)( 78,187)( 79,186)
( 80,185)( 81,192)( 82,191)( 83,190)( 84,189)( 85,196)( 86,195)( 87,194)
( 88,193)( 89,200)( 90,199)( 91,198)( 92,197)( 93,204)( 94,203)( 95,202)
( 96,201)( 97,208)( 98,207)( 99,206)(100,205)(101,212)(102,211)(103,210)
(104,209)(105,216)(106,215)(107,214)(108,213)(217,328)(218,327)(219,326)
(220,325)(221,332)(222,331)(223,330)(224,329)(225,336)(226,335)(227,334)
(228,333)(229,340)(230,339)(231,338)(232,337)(233,344)(234,343)(235,342)
(236,341)(237,348)(238,347)(239,346)(240,345)(241,352)(242,351)(243,350)
(244,349)(245,356)(246,355)(247,354)(248,353)(249,360)(250,359)(251,358)
(252,357)(253,364)(254,363)(255,362)(256,361)(257,368)(258,367)(259,366)
(260,365)(261,372)(262,371)(263,370)(264,369)(265,376)(266,375)(267,374)
(268,373)(269,380)(270,379)(271,378)(272,377)(273,384)(274,383)(275,382)
(276,381)(277,388)(278,387)(279,386)(280,385)(281,392)(282,391)(283,390)
(284,389)(285,396)(286,395)(287,394)(288,393)(289,400)(290,399)(291,398)
(292,397)(293,404)(294,403)(295,402)(296,401)(297,408)(298,407)(299,406)
(300,405)(301,412)(302,411)(303,410)(304,409)(305,416)(306,415)(307,414)
(308,413)(309,420)(310,419)(311,418)(312,417)(313,424)(314,423)(315,422)
(316,421)(317,428)(318,427)(319,426)(320,425)(321,432)(322,431)(323,430)
(324,429);
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*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*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