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