Part of the Atlas of Small Regular Polytopes

Polytope of Type {12,12}

Atlas Canonical Name {12,12}*1728l

▶ Play as a twisty puzzle

Overview

Group
SmallGroup(1728,30242)
Rank
3
Schläfli Type
{12,12}
Vertices, edges, …
72, 432, 72
Order of s0s1s2
12
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

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=<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=<s1*s2*(s1*s0)^2*s1*s2*s1*s0*(s2*s1)^3*s2> 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*s1)^4> of order 3

40 facets

24 vertex figures

P/N, where N=<(s0*s1)^4, s1*s2*(s1*s0)^2*s1*s2*s1*s0*(s2*s1)^3*s2> of order 6

20 facets

12 vertex figures

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

20 facets

12 vertex figures

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

32 facets

12 vertex figures

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

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

References

None.

to this polytope.

Twisty Puzzle