Polytope of Type {216,4}

Play with this polytope as a twisty puzzle

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.
Irregular Quotients (of which this is a minimal cover):
   None.

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*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 ];;
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*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 >;
References : None.
to this polytope

Twisty Puzzle