Part of the Atlas of Small Regular Polytopes

Polytope of Type {36,6}

Atlas Canonical Name {36,6}*1728b

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,30174)
Rank
3
Schläfli Type
{36,6}
Vertices, edges, …
144, 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

12 facets

72 vertex figures

P/N, where N=<(s0*s1)^18> of order 2

16 facets

72 vertex figures

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

12 facets

72 vertex figures

P/N, where N=<(s1*s2)^2> of order 3

8 facets

72 vertex figures

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

6 facets

36 vertex figures

P/N, where N=<s0*(s2*s1)^2*s0*(s1*s2)^2, (s0*s1)^16*s0*s2*s1*s0*s1*s2> of order 4

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

References

None.

to this polytope.

Twisty Puzzle