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}*1728b
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 : 4
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 : {6,36}*864
   3-fold quotients : {6,12}*576b
   4-fold quotients : {6,36}*432a, {6,36}*432c
   6-fold quotients : {6,12}*288a
   8-fold quotients : {6,18}*216a
   9-fold quotients : {6,4}*192b
   12-fold quotients : {2,36}*144, {6,12}*144a, {6,12}*144d
   18-fold quotients : {6,4}*96
   24-fold quotients : {2,18}*72, {6,6}*72a
   36-fold quotients : {2,12}*48, {6,4}*48a, {3,4}*48, {6,4}*48b, {6,4}*48c
   48-fold quotients : {2,9}*36
   72-fold quotients : {3,4}*24, {2,6}*24, {6,2}*24
   108-fold quotients : {2,4}*16
   144-fold quotients : {2,3}*12, {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):
   P/N, where N=<s0*s1*s2*s1*s0*s1*s2*s1> of order 2.
      72 facets:
         72 of {6}*12
      12 vertex figures:
         12 of {36}*72
   P/N, where N=<s0*s2*s1*s2*s1*s0*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=<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*s1*s2> of order 2.
      72 facets:
         72 of {6}*12
      16 vertex figures:
         8 of {36}*72
         8 of {18}*36
   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=<s0*s1*s2*s1*s0*s1*s2*s1, s0*s1*s0*s1*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*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> of order 4.
      36 facets:
         36 of {6}*12
      10 vertex figures:
         8 of {18}*36
         2 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)(  3,232)(  4,231)(  5,238)(  6,237)(  7,240)(  8,239)(  9,234)( 10,233)( 11,236)( 12,235)( 13,218)( 14,217)( 15,220)( 16,219)( 17,226)( 18,225)( 19,228)( 20,227)( 21,222)( 22,221)( 23,224)( 24,223)( 25,250)( 26,249)( 27,252)( 28,251)( 29,246)( 30,245)( 31,248)( 32,247)( 33,242)( 34,241)( 35,244)( 36,243)( 37,266)( 38,265)( 39,268)( 40,267)( 41,274)( 42,273)( 43,276)( 44,275)( 45,270)( 46,269)( 47,272)( 48,271)( 49,254)( 50,253)( 51,256)( 52,255)( 53,262)( 54,261)( 55,264)( 56,263)( 57,258)( 58,257)( 59,260)( 60,259)( 61,286)( 62,285)( 63,288)( 64,287)( 65,282)( 66,281)( 67,284)( 68,283)( 69,278)( 70,277)( 71,280)( 72,279)( 73,302)( 74,301)( 75,304)( 76,303)( 77,310)( 78,309)( 79,312)( 80,311)( 81,306)( 82,305)( 83,308)( 84,307)( 85,290)( 86,289)( 87,292)( 88,291)( 89,298)( 90,297)( 91,300)( 92,299)( 93,294)( 94,293)( 95,296)( 96,295)( 97,322)( 98,321)( 99,324)(100,323)(101,318)(102,317)(103,320)(104,319)(105,314)(106,313)(107,316)(108,315)(109,338)(110,337)(111,340)(112,339)(113,346)(114,345)(115,348)(116,347)(117,342)(118,341)(119,344)(120,343)(121,326)(122,325)(123,328)(124,327)(125,334)(126,333)(127,336)(128,335)(129,330)(130,329)(131,332)(132,331)(133,358)(134,357)(135,360)(136,359)(137,354)(138,353)(139,356)(140,355)(141,350)(142,349)(143,352)(144,351)(145,374)(146,373)(147,376)(148,375)(149,382)(150,381)(151,384)(152,383)(153,378)(154,377)(155,380)(156,379)(157,362)(158,361)(159,364)(160,363)(161,370)(162,369)(163,372)(164,371)(165,366)(166,365)(167,368)(168,367)(169,394)(170,393)(171,396)(172,395)(173,390)(174,389)(175,392)(176,391)(177,386)(178,385)(179,388)(180,387)(181,410)(182,409)(183,412)(184,411)(185,418)(186,417)(187,420)(188,419)(189,414)(190,413)(191,416)(192,415)(193,398)(194,397)(195,400)(196,399)(197,406)(198,405)(199,408)(200,407)(201,402)(202,401)(203,404)(204,403)(205,430)(206,429)(207,432)(208,431)(209,426)(210,425)(211,428)(212,427)(213,422)(214,421)(215,424)(216,423);;
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*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*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 ];;
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)(  3,232)(  4,231)(  5,238)(  6,237)(  7,240)(  8,239)(  9,234)( 10,233)( 11,236)( 12,235)( 13,218)( 14,217)( 15,220)( 16,219)( 17,226)( 18,225)( 19,228)( 20,227)( 21,222)( 22,221)( 23,224)( 24,223)( 25,250)( 26,249)( 27,252)( 28,251)( 29,246)( 30,245)( 31,248)( 32,247)( 33,242)( 34,241)( 35,244)( 36,243)( 37,266)( 38,265)( 39,268)( 40,267)( 41,274)( 42,273)( 43,276)( 44,275)( 45,270)( 46,269)( 47,272)( 48,271)( 49,254)( 50,253)( 51,256)( 52,255)( 53,262)( 54,261)( 55,264)( 56,263)( 57,258)( 58,257)( 59,260)( 60,259)( 61,286)( 62,285)( 63,288)( 64,287)( 65,282)( 66,281)( 67,284)( 68,283)( 69,278)( 70,277)( 71,280)( 72,279)( 73,302)( 74,301)( 75,304)( 76,303)( 77,310)( 78,309)( 79,312)( 80,311)( 81,306)( 82,305)( 83,308)( 84,307)( 85,290)( 86,289)( 87,292)( 88,291)( 89,298)( 90,297)( 91,300)( 92,299)( 93,294)( 94,293)( 95,296)( 96,295)( 97,322)( 98,321)( 99,324)(100,323)(101,318)(102,317)(103,320)(104,319)(105,314)(106,313)(107,316)(108,315)(109,338)(110,337)(111,340)(112,339)(113,346)(114,345)(115,348)(116,347)(117,342)(118,341)(119,344)(120,343)(121,326)(122,325)(123,328)(124,327)(125,334)(126,333)(127,336)(128,335)(129,330)(130,329)(131,332)(132,331)(133,358)(134,357)(135,360)(136,359)(137,354)(138,353)(139,356)(140,355)(141,350)(142,349)(143,352)(144,351)(145,374)(146,373)(147,376)(148,375)(149,382)(150,381)(151,384)(152,383)(153,378)(154,377)(155,380)(156,379)(157,362)(158,361)(159,364)(160,363)(161,370)(162,369)(163,372)(164,371)(165,366)(166,365)(167,368)(168,367)(169,394)(170,393)(171,396)(172,395)(173,390)(174,389)(175,392)(176,391)(177,386)(178,385)(179,388)(180,387)(181,410)(182,409)(183,412)(184,411)(185,418)(186,417)(187,420)(188,419)(189,414)(190,413)(191,416)(192,415)(193,398)(194,397)(195,400)(196,399)(197,406)(198,405)(199,408)(200,407)(201,402)(202,401)(203,404)(204,403)(205,430)(206,429)(207,432)(208,431)(209,426)(210,425)(211,428)(212,427)(213,422)(214,421)(215,424)(216,423);
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*s0*s1*s2*s0*s1*s0*s1*s2*s0*s1*s2*s0*s1, 
s0*s1*s2*s1*s0*s1*s2*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 >;
References : None.
to this polytope

Twisty Puzzle