Polytope of Type {4,54,4}
This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {4,54,4}*1728b
if this polytope has a name.
Group : SmallGroup(1728,11358)
Rank : 4
Schlafli Type : {4,54,4}
Number of vertices, edges, etc : 4, 108, 108, 4
Order of s0s1s2s3 : 108
Order of s0s1s2s3s2s1 : 2
Special Properties :
Universal
Non-Orientable
Flat
Related Polytopes :
Facet
Vertex Figure
Dual
Facet Of :
None in this Atlas
Vertex Figure Of :
None in this Atlas
Quotients (Maximal Quotients in Boldface) :
2-fold quotients : {2,54,4}*864b
3-fold quotients : {4,18,4}*576b
4-fold quotients : {2,27,4}*432
6-fold quotients : {2,18,4}*288b
9-fold quotients : {4,6,4}*192b
12-fold quotients : {2,9,4}*144
18-fold quotients : {2,6,4}*96c
36-fold quotients : {2,3,4}*48
Covers (Minimal Covers in Boldface) :
None in this atlas.
Permutation Representation (GAP) :
s0 := (217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)(224,332)
(225,333)(226,334)(227,335)(228,336)(229,337)(230,338)(231,339)(232,340)
(233,341)(234,342)(235,343)(236,344)(237,345)(238,346)(239,347)(240,348)
(241,349)(242,350)(243,351)(244,352)(245,353)(246,354)(247,355)(248,356)
(249,357)(250,358)(251,359)(252,360)(253,361)(254,362)(255,363)(256,364)
(257,365)(258,366)(259,367)(260,368)(261,369)(262,370)(263,371)(264,372)
(265,373)(266,374)(267,375)(268,376)(269,377)(270,378)(271,379)(272,380)
(273,381)(274,382)(275,383)(276,384)(277,385)(278,386)(279,387)(280,388)
(281,389)(282,390)(283,391)(284,392)(285,393)(286,394)(287,395)(288,396)
(289,397)(290,398)(291,399)(292,400)(293,401)(294,402)(295,403)(296,404)
(297,405)(298,406)(299,407)(300,408)(301,409)(302,410)(303,411)(304,412)
(305,413)(306,414)(307,415)(308,416)(309,417)(310,418)(311,419)(312,420)
(313,421)(314,422)(315,423)(316,424)(317,425)(318,426)(319,427)(320,428)
(321,429)(322,430)(323,431)(324,432);;
s1 := ( 1,217)( 2,218)( 3,220)( 4,219)( 5,225)( 6,226)( 7,228)( 8,227)
( 9,221)( 10,222)( 11,224)( 12,223)( 13,249)( 14,250)( 15,252)( 16,251)
( 17,245)( 18,246)( 19,248)( 20,247)( 21,241)( 22,242)( 23,244)( 24,243)
( 25,237)( 26,238)( 27,240)( 28,239)( 29,233)( 30,234)( 31,236)( 32,235)
( 33,229)( 34,230)( 35,232)( 36,231)( 37,321)( 38,322)( 39,324)( 40,323)
( 41,317)( 42,318)( 43,320)( 44,319)( 45,313)( 46,314)( 47,316)( 48,315)
( 49,309)( 50,310)( 51,312)( 52,311)( 53,305)( 54,306)( 55,308)( 56,307)
( 57,301)( 58,302)( 59,304)( 60,303)( 61,297)( 62,298)( 63,300)( 64,299)
( 65,293)( 66,294)( 67,296)( 68,295)( 69,289)( 70,290)( 71,292)( 72,291)
( 73,285)( 74,286)( 75,288)( 76,287)( 77,281)( 78,282)( 79,284)( 80,283)
( 81,277)( 82,278)( 83,280)( 84,279)( 85,273)( 86,274)( 87,276)( 88,275)
( 89,269)( 90,270)( 91,272)( 92,271)( 93,265)( 94,266)( 95,268)( 96,267)
( 97,261)( 98,262)( 99,264)(100,263)(101,257)(102,258)(103,260)(104,259)
(105,253)(106,254)(107,256)(108,255)(109,325)(110,326)(111,328)(112,327)
(113,333)(114,334)(115,336)(116,335)(117,329)(118,330)(119,332)(120,331)
(121,357)(122,358)(123,360)(124,359)(125,353)(126,354)(127,356)(128,355)
(129,349)(130,350)(131,352)(132,351)(133,345)(134,346)(135,348)(136,347)
(137,341)(138,342)(139,344)(140,343)(141,337)(142,338)(143,340)(144,339)
(145,429)(146,430)(147,432)(148,431)(149,425)(150,426)(151,428)(152,427)
(153,421)(154,422)(155,424)(156,423)(157,417)(158,418)(159,420)(160,419)
(161,413)(162,414)(163,416)(164,415)(165,409)(166,410)(167,412)(168,411)
(169,405)(170,406)(171,408)(172,407)(173,401)(174,402)(175,404)(176,403)
(177,397)(178,398)(179,400)(180,399)(181,393)(182,394)(183,396)(184,395)
(185,389)(186,390)(187,392)(188,391)(189,385)(190,386)(191,388)(192,387)
(193,381)(194,382)(195,384)(196,383)(197,377)(198,378)(199,380)(200,379)
(201,373)(202,374)(203,376)(204,375)(205,369)(206,370)(207,372)(208,371)
(209,365)(210,366)(211,368)(212,367)(213,361)(214,362)(215,364)(216,363);;
s2 := ( 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)(217,253)(218,256)(219,255)(220,254)(221,261)(222,264)
(223,263)(224,262)(225,257)(226,260)(227,259)(228,258)(229,285)(230,288)
(231,287)(232,286)(233,281)(234,284)(235,283)(236,282)(237,277)(238,280)
(239,279)(240,278)(241,273)(242,276)(243,275)(244,274)(245,269)(246,272)
(247,271)(248,270)(249,265)(250,268)(251,267)(252,266)(289,321)(290,324)
(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)(297,313)(298,316)
(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)(306,308)(325,361)
(326,364)(327,363)(328,362)(329,369)(330,372)(331,371)(332,370)(333,365)
(334,368)(335,367)(336,366)(337,393)(338,396)(339,395)(340,394)(341,389)
(342,392)(343,391)(344,390)(345,385)(346,388)(347,387)(348,386)(349,381)
(350,384)(351,383)(352,382)(353,377)(354,380)(355,379)(356,378)(357,373)
(358,376)(359,375)(360,374)(397,429)(398,432)(399,431)(400,430)(401,425)
(402,428)(403,427)(404,426)(405,421)(406,424)(407,423)(408,422)(409,417)
(410,420)(411,419)(412,418)(414,416);;
s3 := ( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)( 15, 16)
( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)( 31, 32)
( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)( 47, 48)
( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)( 63, 64)
( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)( 79, 80)
( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)( 95, 96)
( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)(111,112)
(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)(127,128)
(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)(143,144)
(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)(159,160)
(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)(175,176)
(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)(191,192)
(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)(207,208)
(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)(223,224)
(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)(239,240)
(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)(255,256)
(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)(271,272)
(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)(287,288)
(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)(303,304)
(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)(319,320)
(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)(335,336)
(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)(351,352)
(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)(367,368)
(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)(383,384)
(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)(399,400)
(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)(415,416)
(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)(431,432);;
poly := Group([s0,s1,s2,s3]);;
Finitely Presented Group Representation (GAP) :
F := FreeGroup("s0","s1","s2","s3");;
s0 := F.1;; s1 := F.2;; s2 := F.3;; s3 := F.4;;
rels := [ s0*s0, s1*s1, s2*s2, s3*s3, s0*s2*s0*s2,
s0*s3*s0*s3, s1*s3*s1*s3, s0*s1*s0*s1*s0*s1*s0*s1,
s0*s1*s2*s1*s0*s1*s2*s1, s2*s3*s2*s3*s2*s3*s2*s3,
s3*s2*s1*s3*s2*s3*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*s1*s2*s1*s2 ];;
poly := F / rels;;
Permutation Representation (Magma) :
s0 := Sym(432)!(217,325)(218,326)(219,327)(220,328)(221,329)(222,330)(223,331)
(224,332)(225,333)(226,334)(227,335)(228,336)(229,337)(230,338)(231,339)
(232,340)(233,341)(234,342)(235,343)(236,344)(237,345)(238,346)(239,347)
(240,348)(241,349)(242,350)(243,351)(244,352)(245,353)(246,354)(247,355)
(248,356)(249,357)(250,358)(251,359)(252,360)(253,361)(254,362)(255,363)
(256,364)(257,365)(258,366)(259,367)(260,368)(261,369)(262,370)(263,371)
(264,372)(265,373)(266,374)(267,375)(268,376)(269,377)(270,378)(271,379)
(272,380)(273,381)(274,382)(275,383)(276,384)(277,385)(278,386)(279,387)
(280,388)(281,389)(282,390)(283,391)(284,392)(285,393)(286,394)(287,395)
(288,396)(289,397)(290,398)(291,399)(292,400)(293,401)(294,402)(295,403)
(296,404)(297,405)(298,406)(299,407)(300,408)(301,409)(302,410)(303,411)
(304,412)(305,413)(306,414)(307,415)(308,416)(309,417)(310,418)(311,419)
(312,420)(313,421)(314,422)(315,423)(316,424)(317,425)(318,426)(319,427)
(320,428)(321,429)(322,430)(323,431)(324,432);
s1 := Sym(432)!( 1,217)( 2,218)( 3,220)( 4,219)( 5,225)( 6,226)( 7,228)
( 8,227)( 9,221)( 10,222)( 11,224)( 12,223)( 13,249)( 14,250)( 15,252)
( 16,251)( 17,245)( 18,246)( 19,248)( 20,247)( 21,241)( 22,242)( 23,244)
( 24,243)( 25,237)( 26,238)( 27,240)( 28,239)( 29,233)( 30,234)( 31,236)
( 32,235)( 33,229)( 34,230)( 35,232)( 36,231)( 37,321)( 38,322)( 39,324)
( 40,323)( 41,317)( 42,318)( 43,320)( 44,319)( 45,313)( 46,314)( 47,316)
( 48,315)( 49,309)( 50,310)( 51,312)( 52,311)( 53,305)( 54,306)( 55,308)
( 56,307)( 57,301)( 58,302)( 59,304)( 60,303)( 61,297)( 62,298)( 63,300)
( 64,299)( 65,293)( 66,294)( 67,296)( 68,295)( 69,289)( 70,290)( 71,292)
( 72,291)( 73,285)( 74,286)( 75,288)( 76,287)( 77,281)( 78,282)( 79,284)
( 80,283)( 81,277)( 82,278)( 83,280)( 84,279)( 85,273)( 86,274)( 87,276)
( 88,275)( 89,269)( 90,270)( 91,272)( 92,271)( 93,265)( 94,266)( 95,268)
( 96,267)( 97,261)( 98,262)( 99,264)(100,263)(101,257)(102,258)(103,260)
(104,259)(105,253)(106,254)(107,256)(108,255)(109,325)(110,326)(111,328)
(112,327)(113,333)(114,334)(115,336)(116,335)(117,329)(118,330)(119,332)
(120,331)(121,357)(122,358)(123,360)(124,359)(125,353)(126,354)(127,356)
(128,355)(129,349)(130,350)(131,352)(132,351)(133,345)(134,346)(135,348)
(136,347)(137,341)(138,342)(139,344)(140,343)(141,337)(142,338)(143,340)
(144,339)(145,429)(146,430)(147,432)(148,431)(149,425)(150,426)(151,428)
(152,427)(153,421)(154,422)(155,424)(156,423)(157,417)(158,418)(159,420)
(160,419)(161,413)(162,414)(163,416)(164,415)(165,409)(166,410)(167,412)
(168,411)(169,405)(170,406)(171,408)(172,407)(173,401)(174,402)(175,404)
(176,403)(177,397)(178,398)(179,400)(180,399)(181,393)(182,394)(183,396)
(184,395)(185,389)(186,390)(187,392)(188,391)(189,385)(190,386)(191,388)
(192,387)(193,381)(194,382)(195,384)(196,383)(197,377)(198,378)(199,380)
(200,379)(201,373)(202,374)(203,376)(204,375)(205,369)(206,370)(207,372)
(208,371)(209,365)(210,366)(211,368)(212,367)(213,361)(214,362)(215,364)
(216,363);
s2 := 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)(217,253)(218,256)(219,255)(220,254)(221,261)
(222,264)(223,263)(224,262)(225,257)(226,260)(227,259)(228,258)(229,285)
(230,288)(231,287)(232,286)(233,281)(234,284)(235,283)(236,282)(237,277)
(238,280)(239,279)(240,278)(241,273)(242,276)(243,275)(244,274)(245,269)
(246,272)(247,271)(248,270)(249,265)(250,268)(251,267)(252,266)(289,321)
(290,324)(291,323)(292,322)(293,317)(294,320)(295,319)(296,318)(297,313)
(298,316)(299,315)(300,314)(301,309)(302,312)(303,311)(304,310)(306,308)
(325,361)(326,364)(327,363)(328,362)(329,369)(330,372)(331,371)(332,370)
(333,365)(334,368)(335,367)(336,366)(337,393)(338,396)(339,395)(340,394)
(341,389)(342,392)(343,391)(344,390)(345,385)(346,388)(347,387)(348,386)
(349,381)(350,384)(351,383)(352,382)(353,377)(354,380)(355,379)(356,378)
(357,373)(358,376)(359,375)(360,374)(397,429)(398,432)(399,431)(400,430)
(401,425)(402,428)(403,427)(404,426)(405,421)(406,424)(407,423)(408,422)
(409,417)(410,420)(411,419)(412,418)(414,416);
s3 := Sym(432)!( 1, 2)( 3, 4)( 5, 6)( 7, 8)( 9, 10)( 11, 12)( 13, 14)
( 15, 16)( 17, 18)( 19, 20)( 21, 22)( 23, 24)( 25, 26)( 27, 28)( 29, 30)
( 31, 32)( 33, 34)( 35, 36)( 37, 38)( 39, 40)( 41, 42)( 43, 44)( 45, 46)
( 47, 48)( 49, 50)( 51, 52)( 53, 54)( 55, 56)( 57, 58)( 59, 60)( 61, 62)
( 63, 64)( 65, 66)( 67, 68)( 69, 70)( 71, 72)( 73, 74)( 75, 76)( 77, 78)
( 79, 80)( 81, 82)( 83, 84)( 85, 86)( 87, 88)( 89, 90)( 91, 92)( 93, 94)
( 95, 96)( 97, 98)( 99,100)(101,102)(103,104)(105,106)(107,108)(109,110)
(111,112)(113,114)(115,116)(117,118)(119,120)(121,122)(123,124)(125,126)
(127,128)(129,130)(131,132)(133,134)(135,136)(137,138)(139,140)(141,142)
(143,144)(145,146)(147,148)(149,150)(151,152)(153,154)(155,156)(157,158)
(159,160)(161,162)(163,164)(165,166)(167,168)(169,170)(171,172)(173,174)
(175,176)(177,178)(179,180)(181,182)(183,184)(185,186)(187,188)(189,190)
(191,192)(193,194)(195,196)(197,198)(199,200)(201,202)(203,204)(205,206)
(207,208)(209,210)(211,212)(213,214)(215,216)(217,218)(219,220)(221,222)
(223,224)(225,226)(227,228)(229,230)(231,232)(233,234)(235,236)(237,238)
(239,240)(241,242)(243,244)(245,246)(247,248)(249,250)(251,252)(253,254)
(255,256)(257,258)(259,260)(261,262)(263,264)(265,266)(267,268)(269,270)
(271,272)(273,274)(275,276)(277,278)(279,280)(281,282)(283,284)(285,286)
(287,288)(289,290)(291,292)(293,294)(295,296)(297,298)(299,300)(301,302)
(303,304)(305,306)(307,308)(309,310)(311,312)(313,314)(315,316)(317,318)
(319,320)(321,322)(323,324)(325,326)(327,328)(329,330)(331,332)(333,334)
(335,336)(337,338)(339,340)(341,342)(343,344)(345,346)(347,348)(349,350)
(351,352)(353,354)(355,356)(357,358)(359,360)(361,362)(363,364)(365,366)
(367,368)(369,370)(371,372)(373,374)(375,376)(377,378)(379,380)(381,382)
(383,384)(385,386)(387,388)(389,390)(391,392)(393,394)(395,396)(397,398)
(399,400)(401,402)(403,404)(405,406)(407,408)(409,410)(411,412)(413,414)
(415,416)(417,418)(419,420)(421,422)(423,424)(425,426)(427,428)(429,430)
(431,432);
poly := sub<Sym(432)|s0,s1,s2,s3>;
Finitely Presented Group Representation (Magma) :
poly<s0,s1,s2,s3> := Group< s0,s1,s2,s3 | s0*s0, s1*s1, s2*s2,
s3*s3, s0*s2*s0*s2, s0*s3*s0*s3, s1*s3*s1*s3,
s0*s1*s0*s1*s0*s1*s0*s1, s0*s1*s2*s1*s0*s1*s2*s1,
s2*s3*s2*s3*s2*s3*s2*s3, s3*s2*s1*s3*s2*s3*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*s1*s2*s1*s2 >;
References : None.
to this polytope