Part of the Atlas of Small Regular Polytopes

Polytope of Type {36,12}

Atlas Canonical Name {36,12}*1728c

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,30173)
Rank
3
Schläfli Type
{36,12}
Vertices, edges, …
72, 432, 24
Order of s0s1s2
36
Order of s0s1s2s1
4
Also known as
if this polytope has a name.

Special Properties

  • Compact Hyperbolic Quotient
  • Locally Spherical
  • Orientable

Quotients maximal quotients in bold

2-fold

3-fold

4-fold

6-fold

8-fold

9-fold

12-fold

18-fold

24-fold

36-fold

48-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

Click an entry to reveal its facets and vertex figures.

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2> of order 2

16 facets

36 vertex figures

P/N, where N=<s0*s1*s0*(s2*s1)^4*s0*s2*s1*s2> of order 2

12 facets

36 vertex figures

P/N, where N=<(s0*s1)^2*s2*s1*s0*s1*s2> of order 2

12 facets

36 vertex figures

Representations

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

References

None.

to this polytope.

Twisty Puzzle