Polytope of Type {216,4}

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {216,4}*1728a
Also Known As : {216,4|2}. if this polytope has another name.
Group : SmallGroup(1728,302)
Rank : 3
Schlafli Type : {216,4}
Number of vertices, edges, etc : 216, 432, 4
Order of s₀s₁s₂ : 216
Order of s₀s₁s₂s₁ : 2
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
   Flat
   Self-Petrie
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 : {108,4}*864a, {216,2}*864
   3-fold quotients : {72,4}*576a
   4-fold quotients : {108,2}*432, {54,4}*432a
   6-fold quotients : {36,4}*288a, {72,2}*288
   8-fold quotients : {54,2}*216
   9-fold quotients : {24,4}*192a
   12-fold quotients : {36,2}*144, {18,4}*144a
   16-fold quotients : {27,2}*108
   18-fold quotients : {12,4}*96a, {24,2}*96
   24-fold quotients : {18,2}*72
   27-fold quotients : {8,4}*64a
   36-fold quotients : {12,2}*48, {6,4}*48a
   48-fold quotients : {9,2}*36
   54-fold quotients : {4,4}*32, {8,2}*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)(110,111)(112,117)(113,116)
(114,115)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)(124,129)
(125,128)(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)(146,161)
(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)(163,190)
(164,192)(165,191)(166,198)(167,197)(168,196)(169,195)(170,194)(171,193)
(172,216)(173,215)(174,214)(175,213)(176,212)(177,211)(178,210)(179,209)
(180,208)(181,207)(182,206)(183,205)(184,204)(185,203)(186,202)(187,201)
(188,200)(189,199)(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)
(325,379)(326,381)(327,380)(328,387)(329,386)(330,385)(331,384)(332,383)
(333,382)(334,405)(335,404)(336,403)(337,402)(338,401)(339,400)(340,399)
(341,398)(342,397)(343,396)(344,395)(345,394)(346,393)(347,392)(348,391)
(349,390)(350,389)(351,388)(352,406)(353,408)(354,407)(355,414)(356,413)
(357,412)(358,411)(359,410)(360,409)(361,432)(362,431)(363,430)(364,429)
(365,428)(366,427)(367,426)(368,425)(369,424)(370,423)(371,422)(372,421)
(373,420)(374,419)(375,418)(376,417)(377,416)(378,415);;
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 := (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);;
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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)(110,111)(112,117)
(113,116)(114,115)(118,135)(119,134)(120,133)(121,132)(122,131)(123,130)
(124,129)(125,128)(126,127)(137,138)(139,144)(140,143)(141,142)(145,162)
(146,161)(147,160)(148,159)(149,158)(150,157)(151,156)(152,155)(153,154)
(163,190)(164,192)(165,191)(166,198)(167,197)(168,196)(169,195)(170,194)
(171,193)(172,216)(173,215)(174,214)(175,213)(176,212)(177,211)(178,210)
(179,209)(180,208)(181,207)(182,206)(183,205)(184,204)(185,203)(186,202)
(187,201)(188,200)(189,199)(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)(325,379)(326,381)(327,380)(328,387)(329,386)(330,385)(331,384)
(332,383)(333,382)(334,405)(335,404)(336,403)(337,402)(338,401)(339,400)
(340,399)(341,398)(342,397)(343,396)(344,395)(345,394)(346,393)(347,392)
(348,391)(349,390)(350,389)(351,388)(352,406)(353,408)(354,407)(355,414)
(356,413)(357,412)(358,411)(359,410)(360,409)(361,432)(362,431)(363,430)
(364,429)(365,428)(366,427)(367,426)(368,425)(369,424)(370,423)(371,422)
(372,421)(373,420)(374,419)(375,418)(376,417)(377,416)(378,415);
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)!(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);
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, s0*s1*s2*s1*s0*s1*s2*s1, 
s1*s2*s1*s2*s1*s2*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*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*s0*s1*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