Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,12}

Atlas Canonical Name {12,12}*1728p

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,30326)
Rank
3
Schläfli Type
{12,12}
Vertices, edges, …
72, 432, 72
Order of s0s1s2
6
Order of s0s1s2s1
12
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

16-fold

18-fold

24-fold

36-fold

72-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=<s1*s2*(s1*s0)^2*s1*s2*s1*s0*(s1*s2)^2*s1> of order 2

36 facets

36 vertex figures

P/N, where N=<(s0*s1)^6> of order 2

48 facets

36 vertex figures

P/N, where N=<(s0*s2*s1)^3> of order 2

36 facets

36 vertex figures

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

36 facets

36 vertex figures

P/N, where N=<(s0*s1)^4> of order 3

40 facets

24 vertex figures

P/N, where N=<(s0*s2*s1)^3, (s0*s1)^6> of order 4

24 facets

18 vertex figures

P/N, where N=<(s0*s1)^6, s0*s1*s2*(s1*s0)^4*(s1*s2)^2> of order 4

24 facets

18 vertex figures

P/N, where N=<(s0*s1)^4, s1*s2*(s1*s0)^2*s1*s2*s1*s0*(s1*s2)^2*s1> of order 6

20 facets

12 vertex figures

P/N, where N=<(s0*s1)^4, s0*s2*(s1*s0)^2*s1*s2*s1*s0*(s1*s2)^2> of order 6

24 facets

12 vertex figures

P/N, where N=<(s0*s1)^2> of order 6

32 facets

12 vertex figures

P/N, where N=<(s0*s1)^4, (s0*s2*s1)^3> of order 6

20 facets

12 vertex figures

Representations

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