Part of the Atlas of Small Regular Polytopes

Polytope of Type {4,216}

Atlas Canonical Name {4,216}*1728a

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,302)
Rank
3
Schläfli Type
{4,216}
Vertices, edges, …
4, 432, 216
Order of s0s1s2
216
Order of s0s1s2s1
2
Also known as
{4,216|2}. if this polytope has another name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable
  • Flat

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

16-fold

18-fold

24-fold

27-fold

36-fold

48-fold

54-fold

72-fold

108-fold

144-fold

216-fold

Covers minimal covers in bold

None in this atlas.

Irregular Quotients of which this is a minimal cover

None.

Representations

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

Twisty Puzzle