Polytope of Type {6,36}

Play with this polytope as a twisty puzzle

This page is part of the Atlas of Small Regular Polytopes
Atlas Canonical Name : {6,36}*1728a
if this polytope has a name.
Group : SmallGroup(1728,30174)
Rank : 3
Schlafli Type : {6,36}
Number of vertices, edges, etc : 24, 432, 144
Order of s0s1s2 : 36
Order of s0s1s2s1 : 6
Special Properties :
   Compact Hyperbolic Quotient
   Locally Spherical
   Orientable
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 : {6,18}*864
   3-fold quotients : {6,12}*576a
   4-fold quotients : {6,36}*432b, {6,9}*432
   6-fold quotients : {6,6}*288a
   8-fold quotients : {6,18}*216b
   9-fold quotients : {6,12}*192a
   12-fold quotients : {2,36}*144, {6,12}*144b, {6,3}*144
   16-fold quotients : {6,9}*108
   18-fold quotients : {6,6}*96
   24-fold quotients : {2,18}*72, {6,6}*72b
   36-fold quotients : {2,12}*48, {3,6}*48, {6,3}*48
   48-fold quotients : {2,9}*36, {6,3}*36
   72-fold quotients : {3,3}*24, {2,6}*24
   108-fold quotients : {2,4}*16
   144-fold quotients : {2,3}*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):
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1> of order 2.
      72 facets:
         72 of {6}*12
      12 vertex figures:
         12 of {36}*72
   P/N, where N=<s0*s1*s0*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 2.
      72 facets:
         72 of {6}*12
      12 vertex figures:
         12 of {36}*72
   P/N, where N=<s0*s1*s0*s1> of order 3.
      72 facets:
         36 of {2}*4
         36 of {6}*12
      8 vertex figures:
         8 of {36}*72
   P/N, where N=<s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      6 vertex figures:
         6 of {36}*72
   P/N, where N=<s1*s0*s1*s0*s2*s1*s0*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      6 vertex figures:
         6 of {36}*72
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s0*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      6 vertex figures:
         6 of {36}*72
   P/N, where N=<s0*s1*s0*s2*s1*s0*s1*s0*s2*s1, s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s0*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2*s1*s2> of order 4.
      36 facets:
         36 of {6}*12
      6 vertex figures:
         6 of {36}*72

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

Twisty Puzzle