Polytope of Type {54,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {54,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,11358)
Rank : 3
Schlafli Type : {54,4}
Number of vertices, edges, etc : 216, 432, 16
Order of s0s1s2 : 108
Order of s0s1s2s1 : 4
Special Properties :
Compact Hyperbolic Quotient
Locally Spherical
Orientable
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) :
2-fold quotients : {54,4}*864
3-fold quotients : {18,4}*576b
4-fold quotients : {54,4}*432a, {27,4}*432, {54,4}*432b, {54,4}*432c
6-fold quotients : {18,4}*288
8-fold quotients : {27,4}*216, {54,2}*216
9-fold quotients : {6,4}*192b
12-fold quotients : {18,4}*144a, {9,4}*144, {18,4}*144b, {18,4}*144c
16-fold quotients : {27,2}*108
18-fold quotients : {6,4}*96
24-fold quotients : {9,4}*72, {18,2}*72
36-fold quotients : {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
48-fold quotients : {9,2}*36
72-fold quotients : {3,4}*24, {6,2}*24
108-fold quotients : {2,4}*16
144-fold quotients : {3,2}*12
216-fold quotients : {2,2}*8
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := ( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 13, 33)( 14, 34)( 15, 36)
( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)( 23, 28)
( 24, 27)( 37,105)( 38,106)( 39,108)( 40,107)( 41,101)( 42,102)( 43,104)
( 44,103)( 45, 97)( 46, 98)( 47,100)( 48, 99)( 49, 93)( 50, 94)( 51, 96)
( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)( 59, 88)
( 60, 87)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 77)( 66, 78)( 67, 80)
( 68, 79)( 69, 73)( 70, 74)( 71, 76)( 72, 75)(111,112)(113,117)(114,118)
(115,120)(116,119)(121,141)(122,142)(123,144)(124,143)(125,137)(126,138)
(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,213)(146,214)
(147,216)(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)(154,206)
(155,208)(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)(162,198)
(163,200)(164,199)(165,193)(166,194)(167,196)(168,195)(169,189)(170,190)
(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(177,181)(178,182)
(179,184)(180,183)(219,220)(221,225)(222,226)(223,228)(224,227)(229,249)
(230,250)(231,252)(232,251)(233,245)(234,246)(235,248)(236,247)(237,241)
(238,242)(239,244)(240,243)(253,321)(254,322)(255,324)(256,323)(257,317)
(258,318)(259,320)(260,319)(261,313)(262,314)(263,316)(264,315)(265,309)
(266,310)(267,312)(268,311)(269,305)(270,306)(271,308)(272,307)(273,301)
(274,302)(275,304)(276,303)(277,297)(278,298)(279,300)(280,299)(281,293)
(282,294)(283,296)(284,295)(285,289)(286,290)(287,292)(288,291)(327,328)
(329,333)(330,334)(331,336)(332,335)(337,357)(338,358)(339,360)(340,359)
(341,353)(342,354)(343,356)(344,355)(345,349)(346,350)(347,352)(348,351)
(361,429)(362,430)(363,432)(364,431)(365,425)(366,426)(367,428)(368,427)
(369,421)(370,422)(371,424)(372,423)(373,417)(374,418)(375,420)(376,419)
(377,413)(378,414)(379,416)(380,415)(381,409)(382,410)(383,412)(384,411)
(385,405)(386,406)(387,408)(388,407)(389,401)(390,402)(391,404)(392,403)
(393,397)(394,398)(395,400)(396,399);;
s1 := ( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)( 8, 46)
( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 69)( 14, 72)( 15, 71)( 16, 70)
( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 61)( 22, 64)( 23, 63)( 24, 62)
( 25, 57)( 26, 60)( 27, 59)( 28, 58)( 29, 53)( 30, 56)( 31, 55)( 32, 54)
( 33, 49)( 34, 52)( 35, 51)( 36, 50)( 73,105)( 74,108)( 75,107)( 76,106)
( 77,101)( 78,104)( 79,103)( 80,102)( 81, 97)( 82,100)( 83, 99)( 84, 98)
( 85, 93)( 86, 96)( 87, 95)( 88, 94)( 90, 92)(109,145)(110,148)(111,147)
(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)(119,151)
(120,150)(121,177)(122,180)(123,179)(124,178)(125,173)(126,176)(127,175)
(128,174)(129,169)(130,172)(131,171)(132,170)(133,165)(134,168)(135,167)
(136,166)(137,161)(138,164)(139,163)(140,162)(141,157)(142,160)(143,159)
(144,158)(181,213)(182,216)(183,215)(184,214)(185,209)(186,212)(187,211)
(188,210)(189,205)(190,208)(191,207)(192,206)(193,201)(194,204)(195,203)
(196,202)(198,200)(217,361)(218,364)(219,363)(220,362)(221,369)(222,372)
(223,371)(224,370)(225,365)(226,368)(227,367)(228,366)(229,393)(230,396)
(231,395)(232,394)(233,389)(234,392)(235,391)(236,390)(237,385)(238,388)
(239,387)(240,386)(241,381)(242,384)(243,383)(244,382)(245,377)(246,380)
(247,379)(248,378)(249,373)(250,376)(251,375)(252,374)(253,325)(254,328)
(255,327)(256,326)(257,333)(258,336)(259,335)(260,334)(261,329)(262,332)
(263,331)(264,330)(265,357)(266,360)(267,359)(268,358)(269,353)(270,356)
(271,355)(272,354)(273,349)(274,352)(275,351)(276,350)(277,345)(278,348)
(279,347)(280,346)(281,341)(282,344)(283,343)(284,342)(285,337)(286,340)
(287,339)(288,338)(289,429)(290,432)(291,431)(292,430)(293,425)(294,428)
(295,427)(296,426)(297,421)(298,424)(299,423)(300,422)(301,417)(302,420)
(303,419)(304,418)(305,413)(306,416)(307,415)(308,414)(309,409)(310,412)
(311,411)(312,410)(313,405)(314,408)(315,407)(316,406)(317,401)(318,404)
(319,403)(320,402)(321,397)(322,400)(323,399)(324,398);;
s2 := ( 1,218)( 2,217)( 3,220)( 4,219)( 5,222)( 6,221)( 7,224)( 8,223)
( 9,226)( 10,225)( 11,228)( 12,227)( 13,230)( 14,229)( 15,232)( 16,231)
( 17,234)( 18,233)( 19,236)( 20,235)( 21,238)( 22,237)( 23,240)( 24,239)
( 25,242)( 26,241)( 27,244)( 28,243)( 29,246)( 30,245)( 31,248)( 32,247)
( 33,250)( 34,249)( 35,252)( 36,251)( 37,254)( 38,253)( 39,256)( 40,255)
( 41,258)( 42,257)( 43,260)( 44,259)( 45,262)( 46,261)( 47,264)( 48,263)
( 49,266)( 50,265)( 51,268)( 52,267)( 53,270)( 54,269)( 55,272)( 56,271)
( 57,274)( 58,273)( 59,276)( 60,275)( 61,278)( 62,277)( 63,280)( 64,279)
( 65,282)( 66,281)( 67,284)( 68,283)( 69,286)( 70,285)( 71,288)( 72,287)
( 73,290)( 74,289)( 75,292)( 76,291)( 77,294)( 78,293)( 79,296)( 80,295)
( 81,298)( 82,297)( 83,300)( 84,299)( 85,302)( 86,301)( 87,304)( 88,303)
( 89,306)( 90,305)( 91,308)( 92,307)( 93,310)( 94,309)( 95,312)( 96,311)
( 97,314)( 98,313)( 99,316)(100,315)(101,318)(102,317)(103,320)(104,319)
(105,322)(106,321)(107,324)(108,323)(109,326)(110,325)(111,328)(112,327)
(113,330)(114,329)(115,332)(116,331)(117,334)(118,333)(119,336)(120,335)
(121,338)(122,337)(123,340)(124,339)(125,342)(126,341)(127,344)(128,343)
(129,346)(130,345)(131,348)(132,347)(133,350)(134,349)(135,352)(136,351)
(137,354)(138,353)(139,356)(140,355)(141,358)(142,357)(143,360)(144,359)
(145,362)(146,361)(147,364)(148,363)(149,366)(150,365)(151,368)(152,367)
(153,370)(154,369)(155,372)(156,371)(157,374)(158,373)(159,376)(160,375)
(161,378)(162,377)(163,380)(164,379)(165,382)(166,381)(167,384)(168,383)
(169,386)(170,385)(171,388)(172,387)(173,390)(174,389)(175,392)(176,391)
(177,394)(178,393)(179,396)(180,395)(181,398)(182,397)(183,400)(184,399)
(185,402)(186,401)(187,404)(188,403)(189,406)(190,405)(191,408)(192,407)
(193,410)(194,409)(195,412)(196,411)(197,414)(198,413)(199,416)(200,415)
(201,418)(202,417)(203,420)(204,419)(205,422)(206,421)(207,424)(208,423)
(209,426)(210,425)(211,428)(212,427)(213,430)(214,429)(215,432)(216,431);;
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*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*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*s0*s1 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!( 3, 4)( 5, 9)( 6, 10)( 7, 12)( 8, 11)( 13, 33)( 14, 34)
( 15, 36)( 16, 35)( 17, 29)( 18, 30)( 19, 32)( 20, 31)( 21, 25)( 22, 26)
( 23, 28)( 24, 27)( 37,105)( 38,106)( 39,108)( 40,107)( 41,101)( 42,102)
( 43,104)( 44,103)( 45, 97)( 46, 98)( 47,100)( 48, 99)( 49, 93)( 50, 94)
( 51, 96)( 52, 95)( 53, 89)( 54, 90)( 55, 92)( 56, 91)( 57, 85)( 58, 86)
( 59, 88)( 60, 87)( 61, 81)( 62, 82)( 63, 84)( 64, 83)( 65, 77)( 66, 78)
( 67, 80)( 68, 79)( 69, 73)( 70, 74)( 71, 76)( 72, 75)(111,112)(113,117)
(114,118)(115,120)(116,119)(121,141)(122,142)(123,144)(124,143)(125,137)
(126,138)(127,140)(128,139)(129,133)(130,134)(131,136)(132,135)(145,213)
(146,214)(147,216)(148,215)(149,209)(150,210)(151,212)(152,211)(153,205)
(154,206)(155,208)(156,207)(157,201)(158,202)(159,204)(160,203)(161,197)
(162,198)(163,200)(164,199)(165,193)(166,194)(167,196)(168,195)(169,189)
(170,190)(171,192)(172,191)(173,185)(174,186)(175,188)(176,187)(177,181)
(178,182)(179,184)(180,183)(219,220)(221,225)(222,226)(223,228)(224,227)
(229,249)(230,250)(231,252)(232,251)(233,245)(234,246)(235,248)(236,247)
(237,241)(238,242)(239,244)(240,243)(253,321)(254,322)(255,324)(256,323)
(257,317)(258,318)(259,320)(260,319)(261,313)(262,314)(263,316)(264,315)
(265,309)(266,310)(267,312)(268,311)(269,305)(270,306)(271,308)(272,307)
(273,301)(274,302)(275,304)(276,303)(277,297)(278,298)(279,300)(280,299)
(281,293)(282,294)(283,296)(284,295)(285,289)(286,290)(287,292)(288,291)
(327,328)(329,333)(330,334)(331,336)(332,335)(337,357)(338,358)(339,360)
(340,359)(341,353)(342,354)(343,356)(344,355)(345,349)(346,350)(347,352)
(348,351)(361,429)(362,430)(363,432)(364,431)(365,425)(366,426)(367,428)
(368,427)(369,421)(370,422)(371,424)(372,423)(373,417)(374,418)(375,420)
(376,419)(377,413)(378,414)(379,416)(380,415)(381,409)(382,410)(383,412)
(384,411)(385,405)(386,406)(387,408)(388,407)(389,401)(390,402)(391,404)
(392,403)(393,397)(394,398)(395,400)(396,399);
s1 := Sym(432)!( 1, 37)( 2, 40)( 3, 39)( 4, 38)( 5, 45)( 6, 48)( 7, 47)
( 8, 46)( 9, 41)( 10, 44)( 11, 43)( 12, 42)( 13, 69)( 14, 72)( 15, 71)
( 16, 70)( 17, 65)( 18, 68)( 19, 67)( 20, 66)( 21, 61)( 22, 64)( 23, 63)
( 24, 62)( 25, 57)( 26, 60)( 27, 59)( 28, 58)( 29, 53)( 30, 56)( 31, 55)
( 32, 54)( 33, 49)( 34, 52)( 35, 51)( 36, 50)( 73,105)( 74,108)( 75,107)
( 76,106)( 77,101)( 78,104)( 79,103)( 80,102)( 81, 97)( 82,100)( 83, 99)
( 84, 98)( 85, 93)( 86, 96)( 87, 95)( 88, 94)( 90, 92)(109,145)(110,148)
(111,147)(112,146)(113,153)(114,156)(115,155)(116,154)(117,149)(118,152)
(119,151)(120,150)(121,177)(122,180)(123,179)(124,178)(125,173)(126,176)
(127,175)(128,174)(129,169)(130,172)(131,171)(132,170)(133,165)(134,168)
(135,167)(136,166)(137,161)(138,164)(139,163)(140,162)(141,157)(142,160)
(143,159)(144,158)(181,213)(182,216)(183,215)(184,214)(185,209)(186,212)
(187,211)(188,210)(189,205)(190,208)(191,207)(192,206)(193,201)(194,204)
(195,203)(196,202)(198,200)(217,361)(218,364)(219,363)(220,362)(221,369)
(222,372)(223,371)(224,370)(225,365)(226,368)(227,367)(228,366)(229,393)
(230,396)(231,395)(232,394)(233,389)(234,392)(235,391)(236,390)(237,385)
(238,388)(239,387)(240,386)(241,381)(242,384)(243,383)(244,382)(245,377)
(246,380)(247,379)(248,378)(249,373)(250,376)(251,375)(252,374)(253,325)
(254,328)(255,327)(256,326)(257,333)(258,336)(259,335)(260,334)(261,329)
(262,332)(263,331)(264,330)(265,357)(266,360)(267,359)(268,358)(269,353)
(270,356)(271,355)(272,354)(273,349)(274,352)(275,351)(276,350)(277,345)
(278,348)(279,347)(280,346)(281,341)(282,344)(283,343)(284,342)(285,337)
(286,340)(287,339)(288,338)(289,429)(290,432)(291,431)(292,430)(293,425)
(294,428)(295,427)(296,426)(297,421)(298,424)(299,423)(300,422)(301,417)
(302,420)(303,419)(304,418)(305,413)(306,416)(307,415)(308,414)(309,409)
(310,412)(311,411)(312,410)(313,405)(314,408)(315,407)(316,406)(317,401)
(318,404)(319,403)(320,402)(321,397)(322,400)(323,399)(324,398);
s2 := Sym(432)!( 1,218)( 2,217)( 3,220)( 4,219)( 5,222)( 6,221)( 7,224)
( 8,223)( 9,226)( 10,225)( 11,228)( 12,227)( 13,230)( 14,229)( 15,232)
( 16,231)( 17,234)( 18,233)( 19,236)( 20,235)( 21,238)( 22,237)( 23,240)
( 24,239)( 25,242)( 26,241)( 27,244)( 28,243)( 29,246)( 30,245)( 31,248)
( 32,247)( 33,250)( 34,249)( 35,252)( 36,251)( 37,254)( 38,253)( 39,256)
( 40,255)( 41,258)( 42,257)( 43,260)( 44,259)( 45,262)( 46,261)( 47,264)
( 48,263)( 49,266)( 50,265)( 51,268)( 52,267)( 53,270)( 54,269)( 55,272)
( 56,271)( 57,274)( 58,273)( 59,276)( 60,275)( 61,278)( 62,277)( 63,280)
( 64,279)( 65,282)( 66,281)( 67,284)( 68,283)( 69,286)( 70,285)( 71,288)
( 72,287)( 73,290)( 74,289)( 75,292)( 76,291)( 77,294)( 78,293)( 79,296)
( 80,295)( 81,298)( 82,297)( 83,300)( 84,299)( 85,302)( 86,301)( 87,304)
( 88,303)( 89,306)( 90,305)( 91,308)( 92,307)( 93,310)( 94,309)( 95,312)
( 96,311)( 97,314)( 98,313)( 99,316)(100,315)(101,318)(102,317)(103,320)
(104,319)(105,322)(106,321)(107,324)(108,323)(109,326)(110,325)(111,328)
(112,327)(113,330)(114,329)(115,332)(116,331)(117,334)(118,333)(119,336)
(120,335)(121,338)(122,337)(123,340)(124,339)(125,342)(126,341)(127,344)
(128,343)(129,346)(130,345)(131,348)(132,347)(133,350)(134,349)(135,352)
(136,351)(137,354)(138,353)(139,356)(140,355)(141,358)(142,357)(143,360)
(144,359)(145,362)(146,361)(147,364)(148,363)(149,366)(150,365)(151,368)
(152,367)(153,370)(154,369)(155,372)(156,371)(157,374)(158,373)(159,376)
(160,375)(161,378)(162,377)(163,380)(164,379)(165,382)(166,381)(167,384)
(168,383)(169,386)(170,385)(171,388)(172,387)(173,390)(174,389)(175,392)
(176,391)(177,394)(178,393)(179,396)(180,395)(181,398)(182,397)(183,400)
(184,399)(185,402)(186,401)(187,404)(188,403)(189,406)(190,405)(191,408)
(192,407)(193,410)(194,409)(195,412)(196,411)(197,414)(198,413)(199,416)
(200,415)(201,418)(202,417)(203,420)(204,419)(205,422)(206,421)(207,424)
(208,423)(209,426)(210,425)(211,428)(212,427)(213,430)(214,429)(215,432)
(216,431);
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*s2*s1*s2*s1*s0*s1*s0*s1*s0*s2*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*s0*s1 >;
References : None.
to this polytope